Atomic Physics

Atomic PhysicsAtomic Physics

Step PotentialStep Potential

Consider a particle of energyConsider a particle of energy EE moving in region in which the moving in region in which the potential energy is the step functionpotential energy is the step function

U(x) = 0,U(x) = 0, x<0x<0

U(x) = VU(x) = V00,, x>0 x>0

What happened when What happened when

a particle moving from a particle moving from

left to right encountersleft to right encounters

the step?the step?

The classical answer isThe classical answer is

simple: to the left of the simple: to the left of the

step, the particle movesstep, the particle moves

with a speed with a speed v = v = √2E/m√2E/m

Step PotentialStep PotentialAtAt x =0x =0, , an impulsive force an impulsive force act on the particle. If the act on the particle. If the initial energyinitial energy EE is less thanis less than VV00, , the particle will be the particle will be

turned around and will then turned around and will then move to the left at its move to the left at its original speed; that is, the original speed; that is, the particle will be reflected by particle will be reflected by the step. Ifthe step. If EE is greater than is greater than VV00, the particle will continue , the particle will continue

to move to the right but with to move to the right but with reduced speed given byreduced speed given by

v = v = √2(E – U√2(E – U00)/m)/m

Step PotentialStep Potential

We can picture this classical problem as a We can picture this classical problem as a ball rolling along a level surface and coming to a ball rolling along a level surface and coming to a steep hill of heightsteep hill of height hh given bygiven by mgh=Vmgh=V00. .

If the initial kinetic energy of the ball is less If the initial kinetic energy of the ball is less thanthan mghmgh, the ball will roll part way up the hill and , the ball will roll part way up the hill and then back down and to the left along the lower then back down and to the left along the lower surface at it original speed. Ifsurface at it original speed. If EE is greater thanis greater than mghmgh, the ball will roll up the hill and proceed to the , the ball will roll up the hill and proceed to the right at a lesser speed. right at a lesser speed.

The quantum mechanical result is similar whenThe quantum mechanical result is similar when EE is is less thanless than VV00.. If If E<VE<V00 the wave function does not go to the wave function does not go to

zero at zero at x=0x=0 but rather decays exponentially. The wave but rather decays exponentially. The wave penetrates slightly into the classically forbidden regionpenetrates slightly into the classically forbidden region x>0x>0, but it is eventually completely reflected. , but it is eventually completely reflected.

Step PotentialStep PotentialThis problem is somewhat similar to that of This problem is somewhat similar to that of

total internal reflection in optics. total internal reflection in optics.

ForFor E>VE>V00, , the quantum mechanical result differs the quantum mechanical result differs

from the classical result. Atfrom the classical result. At x=0x=0, the wavelength , the wavelength changes from changes from

λλ11=h/p=h/p11 = h/√2mE = h/√2mE

toto

λλ22=h/p=h/p22 = h/√2m(E-V = h/√2m(E-V00).).

When the wavelength changes suddenly, part of When the wavelength changes suddenly, part of

the wave is reflected and part of the wave is the wave is reflected and part of the wave is transmitted.transmitted.

Reflection CoefficientReflection Coefficient

Since a motion of an electron (or other Since a motion of an electron (or other

particle) is governed by a wave equation, the particle) is governed by a wave equation, the

electron sometimes will be electron sometimes will be transmitted transmitted and and

sometimes will be sometimes will be reflectedreflected. .

The probabilities of reflection and The probabilities of reflection and

transmission can be calculated by solving the transmission can be calculated by solving the

SchrSchröödinger equationdinger equation in each region of space in each region of space

and comparing the amplitudes of transmitted and comparing the amplitudes of transmitted

waves and reflected waves with that of the waves and reflected waves with that of the

incident wave. incident wave.

Reflection CoefficientReflection Coefficient

This calculation and its result are similar to finding This calculation and its result are similar to finding

the fraction of light reflected from the air-glass interface. Ifthe fraction of light reflected from the air-glass interface. If

RR is the probability of reflection, called the is the probability of reflection, called the reflection reflection

coefficientcoefficient, this calculation gives:, this calculation gives:

wherewhere kk11 is the wave number for the incident wave andis the wave number for the incident wave and kk22

is the wave number for the transmitted wave. is the wave number for the transmitted wave.

221

221

)(

)(

kk

kkR

Transmission CoefficientTransmission Coefficient The result is the same as the result in optics for The result is the same as the result in optics for the reflection of light at normal incidence from the the reflection of light at normal incidence from the boundary between two media having different boundary between two media having different indexes indexes of refractionof refraction nn..

The probability of transmissionThe probability of transmission TT, called the , called the transmission coefficienttransmission coefficient, can be calculated from the , can be calculated from the reflection coefficient, since the probability of reflection coefficient, since the probability of transmission plus the probability of reflection must transmission plus the probability of reflection must equal 1:equal 1:

T + R = 1 T + R = 1 In the quantum mechanics, a localized particle is In the quantum mechanics, a localized particle is

represented by the wave packet, which has arepresented by the wave packet, which has a maximum maximum at the most probable position of the particle. at the most probable position of the particle.

Time development of a one dimensional wave packet Time development of a one dimensional wave packet representing a particle incident on a step potential forrepresenting a particle incident on a step potential for E>VE>V00. .

The position of a classical particle is indicated by the dot. Note The position of a classical particle is indicated by the dot. Note that part of the packet is transmitted and part is reflected.that part of the packet is transmitted and part is reflected.

Reflection coefficientReflection coefficient RR and transmission coefficientand transmission coefficient TT for a for a potential steppotential step VV00 high versus energyhigh versus energy EE (in units(in units VV00).).

A particle of energyA particle of energy EE00 traveling in a region in traveling in a region in

which the potential energy is zero is incident on which the potential energy is zero is incident on a potential barrier of heighta potential barrier of height VV00=0.2E=0.2E00. . Find the Find the

probability thatprobability that the particle will be reflected. the particle will be reflected.

Lets consider a rectangular potential barrier of heightLets consider a rectangular potential barrier of height VV00 and and

withwith aa given by:given by:U(x) = 0,U(x) = 0, x<0x<0U(x) = VU(x) = V00,, 0<x<a0<x<a

U(x) = 0,U(x) = 0, x>ax>a

Barrier PotentialBarrier PotentialWe consider a particle of We consider a particle of energyenergy EE , which is slightly , which is slightly less thanless than VV00, that is incident , that is incident

on the barrier from the left. on the barrier from the left. Classically, the particle Classically, the particle would always be reflected. would always be reflected. However, a wave incident However, a wave incident from the left does not from the left does not decrease immediately to decrease immediately to zero at the barrier, but itzero at the barrier, but itwill instead decay exponentially in the classically forbidden will instead decay exponentially in the classically forbidden regionregion 0<x<a0<x<a. Upon reaching the far wall. Upon reaching the far wall of the barrierof the barrier (x=a),(x=a), the wave function must join smoothlythe wave function must join smoothly to a sinusoidal to a sinusoidal wave function to the right of barrier.wave function to the right of barrier.

Barrier PotentialBarrier PotentialIf we have a beam of particle incident from left, all with the If we have a beam of particle incident from left, all with the

same energysame energy E<VE<V00, the general solution of the wave equation are, , the general solution of the wave equation are,

following the example for a potential step,following the example for a potential step,

wherewhere kk11 =√2mE/ħ =√2mE/ħ and and αα = √2m(V = √2m(V00-E)/ħ-E)/ħ

This implies that there is some probability of the particle (which is This implies that there is some probability of the particle (which is represented by the wave function) being found on the far side of represented by the wave function) being found on the far side of the barrier even though, classically, it should never pass through the barrier even though, classically, it should never pass through the barrier. the barrier.

axGeFex

axDeCex

xBeAex

xikxik

xx

xikxik

11

11

)(

0)(

0)(

3

2

1

Barrier PotentialBarrier Potential

For the case in which the quantityFor the case in which the quantity

ααa = √2maa = √2ma22(V(V00 – E)/ħ – E)/ħ22

is much greater thanis much greater than 1 1, the transmission coefficient , the transmission coefficient is proportionalis proportional toto ee-2-2ααaa, with, with

αα = √2m(V = √2m(V00 – E)/ħ – E)/ħ22

The probability of penetration of the barrier thus The probability of penetration of the barrier thus decreases exponentially with the barrier thicknessdecreases exponentially with the barrier thickness aa and with the square root of the relative barrier and with the square root of the relative barrier heightheight (V(V00-E)-E). This phenomenon is called barrier . This phenomenon is called barrier

penetration or tunneling. The relative probability of penetration or tunneling. The relative probability of its occurrence in any given situation is given by the its occurrence in any given situation is given by the transmission coefficient. transmission coefficient.

A wave packet representing a particle incident on two A wave packet representing a particle incident on two barriers of height just slightly greater than the energy of the barriers of height just slightly greater than the energy of the particle. At each encounter, part of the packet is transmitted particle. At each encounter, part of the packet is transmitted and part reflected, resulting in part of the packet being and part reflected, resulting in part of the packet being trapped between the barriers from same time. trapped between the barriers from same time.

A A 30-eV30-eV electron is incident on a square barrier of electron is incident on a square barrier of height height 40 eV40 eV. What is the probability that the electron will . What is the probability that the electron will tunnel through the barrier if its width is (a) tunnel through the barrier if its width is (a) 1.0 nm1.0 nm? ?

(b) (b) 0.1nm0.1nm??

The penetration of the barrier is not unique to quantum The penetration of the barrier is not unique to quantum mechanics. When light is totally reflected from the glass-air mechanics. When light is totally reflected from the glass-air interface, the light wave can penetrate the air barrier if a interface, the light wave can penetrate the air barrier if a second peace of glass is brought within a few wavelengths of second peace of glass is brought within a few wavelengths of the first, even when the angle of incidence in the first prism is the first, even when the angle of incidence in the first prism is greater than the critical angle. This effect can be demonstrated greater than the critical angle. This effect can be demonstrated with a laser beam and two with a laser beam and two 4545°° prisms. prisms.

αα- Decay- Decay

The theory of barrier penetration was used by The theory of barrier penetration was used by George GamovGeorge Gamov in 1928 to in 1928 to

explain the enormous variation of the half-lives forexplain the enormous variation of the half-lives for αα decay of radioactive decay of radioactive

nuclei. Potential well shown on the diagram for annuclei. Potential well shown on the diagram for an αα particle in a particle in a

radioactive nucleus approximately describes a strong attractive force radioactive nucleus approximately describes a strong attractive force

whenwhen rr is less than the nuclear radiusis less than the nuclear radius RR.. Outside the nucleus the strong Outside the nucleus the strong

nuclear force is negligible, and the potential is given by the nuclear force is negligible, and the potential is given by the Coulomb’s Coulomb’s

lawlaw,, U(r) = +k(2e)(Ze)/rU(r) = +k(2e)(Ze)/r, , wherewhere ZZ is the nuclear charge andis the nuclear charge and 2e2e is the is the

charge of charge of αα particle.particle.

αα- Decay- Decay

An An αα-particle inside the nucleus oscillates back and forth, being reflected -particle inside the nucleus oscillates back and forth, being reflected

at the barrier at at the barrier at RR. Because of its wave properties, when the . Because of its wave properties, when the αα-particle -particle

hits the barrier there is a small chance that it will penetrate and appear hits the barrier there is a small chance that it will penetrate and appear

outside the well at outside the well at r = rr = r00. The wave function is similar to that for a square . The wave function is similar to that for a square

barrier potential. barrier potential.

The probability that an The probability that an αα-particle will tunnel through the -particle will tunnel through the barrier is given by barrier is given by

which is a very small number, i.e., the which is a very small number, i.e., the αα particle is usually particle is usually reflected. The number of times per second reflected. The number of times per second NN that the that the αα particle approaches the barrier is given byparticle approaches the barrier is given by

aEVm

eT)(22 0

R

vN

2

where where vv equals the particle’s speed inside the nucleus. equals the particle’s speed inside the nucleus.

The decay rate, or the probability per second that the nucleus The decay rate, or the probability per second that the nucleus will emit an will emit an αα particle, which is also the reciprocal of the mean particle, which is also the reciprocal of the mean life time , is given bylife time , is given by

aEVm

eR

vratedecay

)(22 0

2

1

The decay rate for emission ofThe decay rate for emission of αα particles from radioactive particles from radioactive nuclei ofnuclei of PoPo212212. . The solid curve is the prediction of equationThe solid curve is the prediction of equation

The points are the experimental results.The points are the experimental results.

aEVm

eR

vratedecay

)(22 0

2

1

Applications of TunnelingApplications of Tunneling

• NanotechnologyNanotechnology refers to the design and application of refers to the design and application of devices having dimensions ranging from devices having dimensions ranging from 11 to to 100 nm100 nm

• Nanotechnology uses the idea of trapping particles in Nanotechnology uses the idea of trapping particles in potential wellspotential wells

• One area of nanotechnology of interest to researchers is the One area of nanotechnology of interest to researchers is the quantum dotquantum dot– A quantum dot is a small region that is grown in a silicon A quantum dot is a small region that is grown in a silicon

crystal that acts as a potential wellcrystal that acts as a potential well

• Nuclear fusionNuclear fusion– Protons can tunnel through the barrier caused by their Protons can tunnel through the barrier caused by their

mutual electrostatic repulsionmutual electrostatic repulsion

Resonant Tunneling DeviceResonant Tunneling Device

• Electrons travel in the gallium arsenide Electrons travel in the gallium arsenide semiconductor semiconductor

• They strike the barrier of the quantum dot from the They strike the barrier of the quantum dot from the leftleft

• The electrons can tunnel through the barrier and The electrons can tunnel through the barrier and produce a current in the deviceproduce a current in the device

Scanning Tunneling MicroscopeScanning Tunneling Microscope

• An electrically conducting An electrically conducting probe with a very sharp probe with a very sharp edge is brought near the edge is brought near the surface to be studiedsurface to be studied

• The empty space The empty space between the tip and the between the tip and the surface represents the surface represents the “barrier”“barrier”

• The tip and the surface The tip and the surface are two walls of the are two walls of the “potential well”“potential well”


• The The STMSTM allows allows highly detailed highly detailed images of surfaces images of surfaces with resolutions with resolutions comparable to the comparable to the size of a single atomsize of a single atom

• At right is the surface At right is the surface of graphite “viewed” of graphite “viewed” with the with the STMSTM


• The The STMSTM is very sensitive to the distance from is very sensitive to the distance from the tip to the surfacethe tip to the surface– This is the thickness of the barrierThis is the thickness of the barrier

• STMSTM has one very serious limitation has one very serious limitation– Its operation is dependent on the electrical Its operation is dependent on the electrical

conductivity of the sample and the tipconductivity of the sample and the tip– Most materials are not electrically conductive at their Most materials are not electrically conductive at their

surfacessurfaces– The atomic force microscope (The atomic force microscope (AFMAFM) overcomes this ) overcomes this

limitation by tracking the sample surface maintaining limitation by tracking the sample surface maintaining a constant interatomic force between the atoms on a constant interatomic force between the atoms on the scanner tip and the sample’s surface atoms. the scanner tip and the sample’s surface atoms.

SUMMARYSUMMARY1. Time-independent Schr1. Time-independent Schrödinger equation:ödinger equation:

2.In the simple harmonic oscillator:2.In the simple harmonic oscillator:

the ground wave function is given:the ground wave function is given:

where where AA00 is the normalization constant andis the normalization constant and a=ma=mωω00/2ħ/2ħ. .

3. In a finite square well of height3. In a finite square well of height VV00, , there are only a finite there are only a finite number of allowed energies. number of allowed energies.

)()()()(

2 2

22

xExxUdx

xd

m

02

1

nEn

2

)(0axAex

SUMMARYSUMMARY

4.Reflection and barrier penetration:4.Reflection and barrier penetration:

When the potentials changes abruptly over a When the potentials changes abruptly over a small distance, a particle may be reflected even small distance, a particle may be reflected even thoughthough E>U(x)E>U(x).. A particle may penetrate a A particle may penetrate a region in whichregion in which E<U(x)E<U(x).. Reflection and Reflection and penetration of electron waves are similar for penetration of electron waves are similar for those for other kinds of waves. those for other kinds of waves.

The SchrThe Schrödinger Equation in Three Dimensionsödinger Equation in Three Dimensions

The one-dimensional time-independent SchrThe one-dimensional time-independent Schrödinger ödinger equationequation

(1)

is easily extended to three dimensions. In rectangular is easily extended to three dimensions. In rectangular coordinates, it iscoordinates, it is

(2)

where the wave functionwhere the wave function ψψ and the potential energyand the potential energy UU are are generally functions of all three coordinates ,generally functions of all three coordinates , xx, , yy,, and and zz. .

)()()()(

2 2

22

xExxUdx

xd

m

EU

dz

d

dy

d

dx

d

m 2

2

2

2

2

22

2

The SchrThe Schrödinger Equation in Three ödinger Equation in Three DimensionsDimensions

To illustrate some of the features of To illustrate some of the features of problems in three dimensions, we consider a problems in three dimensions, we consider a particle in three-dimensional infinity square well particle in three-dimensional infinity square well given by given by U(x,y,z)U(x,y,z) =0=0 forfor 0<x<L0<x<L,, 0<y<L0<y<L,, and and 0<z<L0<z<L. .

Outside this cubical region,Outside this cubical region, U(x,y,z)=∞.U(x,y,z)=∞. For For this problem, the wave function must be zero at this problem, the wave function must be zero at the edges of the well. the edges of the well.


The standard method for solving this partial differential The standard method for solving this partial differential

equation is guess the form of the solution using the probability. equation is guess the form of the solution using the probability. For a one-dimensional box along theFor a one-dimensional box along the xx axis, we have found the axis, we have found the probability that the particle is in the regionprobability that the particle is in the region dxdx at at xx to beto be

AA1122sinsin22(k(k11x)dxx)dx (3)(3)

wherewhere AA11 is the normalization constant, andis the normalization constant, and kk11=n=nππ/L/L is the is the

wave number. Similarly, for a box alongwave number. Similarly, for a box along yy axis, the probability axis, the probability of a particle being in a regionof a particle being in a region dydy atat yy is is

AA2222sinsin22(k(k22y)dyy)dy (4)(4)

The probability of two independent events occurring is The probability of two independent events occurring is the product of probabilities of each event occurring. the product of probabilities of each event occurring.


So, the probability of a particle being in regionSo, the probability of a particle being in region dxdx at at xx and in regionand in region dydy atat yy isis

The probability of a particle being in the regionThe probability of a particle being in the region dx, dydx, dy, , andand dzdz is is ψψ22(x,y,x)dxdydz(x,y,x)dxdydz, where, where ψψ(x,y,z)(x,y,z) is the solution of is the solution of equationequation

(2)(2)

EzyxU

dz

d

dy

d

dx

d

m),,(

2 2

2

2

2

2

22

dxdyykAxkAdyykAdxxkA )(sin)(sin)(sin)(sin 222

2122

1222

2122

1


EzyxU

dz

d

dy

d

dx

d

m),,(

2 2

2

2

2

2

22

)(2

23

22

21

2

kkkm

E

m

pppE zyx

2

222

dxdydzzkykxkAzyx )sin()sin()sin(),,( 321

The solution is of the formThe solution is of the form

where the constantwhere the constant AA is determined by normalization. is determined by normalization. Inserting this solution in the equation (2), we obtain for the Inserting this solution in the equation (2), we obtain for the energy:energy:

which is equivalent to:which is equivalent to:

with with ppxx=ħk=ħk11 and so on.and so on.

(2)


The wave function will be zero atThe wave function will be zero at x=Lx=L ifif kk11=n=n11ππ/L/L,, wherewhere nn11 is the integer. Similarly, the wave function will be is the integer. Similarly, the wave function will be zero atzero at y=Ly=L if if kk22=n=n22ππ/L/L, and the wave function will be zero , and the wave function will be zero atat z=Lz=L ifif kk33=n=n33ππ/L/L. It is also zero at. It is also zero at x=0x=0,, y=0y=0,, and and z=0z=0. The . The energy is thus quantized to the valuesenergy is thus quantized to the values

wherewhere nn11,, nn22,, andand nn33 are integers andare integers and EE11 is the ground-state is the ground-state

energy of the one dimensional well.energy of the one dimensional well.The lowest energy state (the ground state) for the The lowest energy state (the ground state) for the

cubical well occurs whencubical well occurs when nn1122=n=n22

22=n=n3322=1=1 and has the valueand has the value

)()(2

23

22

211

23

22

212

22

321nnnEnnn

mLE nnn

12

22

1,1,1 32

3E

mLE


The first excited energy level can be obtained in three The first excited energy level can be obtained in three different ways:different ways:

1) n1) n11=2, n=2, n22=n=n33=1; 2) n=1; 2) n22=2, n=2, n11=n=n33=1; 3) n=1; 3) n33=2, n=2, n11=n=n22=1=1. .

Each way has a different wave function . For example, the Each way has a different wave function . For example, the wave function forwave function for nn11=2,=2, nn22=n=n33=1=1 is:is:

There are thus three different quantum states as There are thus three different quantum states as described by three different wave functions corresponding to described by three different wave functions corresponding to the same energy level. The energy level with more than one the same energy level. The energy level with more than one wave function are associate is said to be wave function are associate is said to be degeneratedegenerate. In this . In this case, there is threefold case, there is threefold degeneracydegeneracy..

L

z

L

y

L

xA

sinsin

2sin1,1,2


Degeneracy is related to the spatial symmetry Degeneracy is related to the spatial symmetry of the system. If, for example, we consider a of the system. If, for example, we consider a noncubic well, wherenoncubic well, where U=0U=0 forfor 0<x<L0<x<L11,, 0<y<L0<y<L22, and, and

0<z<L0<z<L33, , the boundary conditions at the edges would the boundary conditions at the edges would

lead to the quantum conditionslead to the quantum conditions kk11LL11=n=n11ππ,, kk22LL22=n=n22ππ, ,

andand kk33LL33=n=n33ππ and the total energy would be:and the total energy would be:

23

23

22

22

21

21

22

2321 L

n

L

n

L

n

mE nnn

This energy level are not degenerate ifThis energy level are not degenerate if LL11,, LL22, and, and LL33 are all are all

different.different.

Figure shows the energy levels for the ground state and Figure shows the energy levels for the ground state and first two excited levels for an infinity cubic well in which the first two excited levels for an infinity cubic well in which the excited states are degenerated and for a noncubic infinity well in excited states are degenerated and for a noncubic infinity well in whichwhich LL11,, LL22,, and and LL33 are all slightly different so that the excited are all slightly different so that the excited

levels are slightly split apart and the degeneracy is removed.levels are slightly split apart and the degeneracy is removed.

The ground state is the state where the The ground state is the state where the

quantum numbersquantum numbers nn11,, nn22, and, and nn33 are all equal to are all equal to 11. .

Non of the three quantum numbers can be zero. If Non of the three quantum numbers can be zero. If any one ofany one of nn11, , nn22, and, and nn33 were were zerozero, the , the

corresponding wave numbercorresponding wave number kk would also equal to would also equal to zerozero and corresponding wave function would equal and corresponding wave function would equal to zero for all valuesto zero for all values ofof x,yx,y,, and and zz. .

The Degenerate StatesThe Degenerate States

Example 1Example 1..

A particle is in three-dimensional box withA particle is in three-dimensional box with LL33=L=L22=2L=2L11.. Give the quantum numbers Give the quantum numbers nn11,,

nn22,, and and nn33 that correspond to the thirteen that correspond to the thirteen

quantum states of this box that have the quantum states of this box that have the lowest energies. lowest energies.

Example 2.Example 2.

Write the degenerate wave function for the Write the degenerate wave function for the fourth and fifth excited states (level fourth and fifth excited states (level 55 and and 66) from the Example1.) from the Example1.

The SchrThe Schrödinger Equation for Two Identical Particlesödinger Equation for Two Identical Particles

Thus far our quantum mechanical Thus far our quantum mechanical

consideration was limited to situation in which a consideration was limited to situation in which a single particle moves in some force field single particle moves in some force field characterized by a potential energy functioncharacterized by a potential energy function UU. .

The most important physical problem of this The most important physical problem of this type is the hydrogen atom, in which a single type is the hydrogen atom, in which a single electron moves in the electron moves in the CoulombCoulomb potential of the potential of the proton nucleus. proton nucleus.


This problem is actually a two-body problem, This problem is actually a two-body problem,

since the proton also moves in the field of electron. since the proton also moves in the field of electron. However, the motion of the much more massive However, the motion of the much more massive proton requires only a very small correction to the proton requires only a very small correction to the energy of the atom that is easily made in both energy of the atom that is easily made in both classical and quantum mechanics. classical and quantum mechanics.

When we consider more complicated When we consider more complicated problems, such as the helium atom, we must apply problems, such as the helium atom, we must apply the quantum mechanics to two or more electrons the quantum mechanics to two or more electrons moving in an external field.moving in an external field.


The interaction of two electrons with each The interaction of two electrons with each other is electromagnetic and is essentially the other is electromagnetic and is essentially the same, that the classical interaction of two charged same, that the classical interaction of two charged particles. particles.

The SchrThe Schrödinger equation for an atom with ödinger equation for an atom with two or more electrons cannot be solved exactly, so two or more electrons cannot be solved exactly, so approximation method must be used. This is not approximation method must be used. This is not very different from classical problem with three or very different from classical problem with three or more particles, however, the complications arising more particles, however, the complications arising from the identity of electrons. from the identity of electrons.


There are due to the fact that it is impossible to There are due to the fact that it is impossible to

keep track of which electron is which.keep track of which electron is which.

Classically, identical particles can be identified Classically, identical particles can be identified by their position, which can be determined with by their position, which can be determined with unlimited accuracy. unlimited accuracy.

This is impossible quantum mechanically This is impossible quantum mechanically because of the uncertainty principle. because of the uncertainty principle.


The undistinguishability of identical particles The undistinguishability of identical particles has important consequence. For instance, has important consequence. For instance, consider the very simple case of two identical, consider the very simple case of two identical, noninteracting particles in one-dimensional infinity noninteracting particles in one-dimensional infinity square well. square well.

The time independent SchrThe time independent Schrödinger equation ödinger equation for two particles, each massfor two particles, each mass mm, is, is

where where xx11 andand x x22 are the coordinates of the are the coordinates of the

two particles. two particles.

)()(),(

2

),(

2 212122

2122

21

2122

xxExxUdx

xxd

mdx

xxd

m


If the particles interact, the potential energyIf the particles interact, the potential energy UU contains terms with bothcontains terms with both xx11 andand xx22 that can not be that can not be separated. For example, the electrostatic repulsion of two separated. For example, the electrostatic repulsion of two electrons in one dimension is represented by potential electrons in one dimension is represented by potential energyenergy keke22/(x/(x22-x-x11).).

However if the particles do not interact, as we However if the particles do not interact, as we assuming here, we can writeassuming here, we can write U = U(xU = U(x11) + U(x) + U(x22).).

For the infinity square well, we need only solve the For the infinity square well, we need only solve the ShrShröödinger equation inside the well where dinger equation inside the well where U=0U=0, and , and require that the wave function be zero at the walls of the require that the wave function be zero at the walls of the well. well.


WithWith U=0U=0, equation, equation

looks just like the expression for a two-dimensionallooks just like the expression for a two-dimensionalwellwell

with nowith no zz and withand with yy replaced byreplaced by xx22..

)()(),(

2

),(

2 212122

2122

21

2122

xxExxUdx

xxd

mdx

xxd

m

EU

dz

d

dy

d

dx

d

m 2

2

2

2

2

22

2


Solution of this equation can be written in Solution of this equation can be written in the formthe form

ΨΨn,mn,m= = ψψnn(x(x11))ψψmm(x(x22))

where where ψψnn andand ψψmm are the single particle wave are the single particle wave

function for a particle in the infinity well andfunction for a particle in the infinity well and nn and and mm are the quantum numbers of particles are the quantum numbers of particles 11 and and 22. . For example, forFor example, for n=1n=1 andand m=2m=2 the wave function the wave function isis

EU

dz

d

dy

d

dx

d

m 2

2

2

2

2

22

2

L

x

L

xA 21

2,1

2sinsin


The probability of finding particle The probability of finding particle 11 inin dxdx11 and particle and particle 22 in in

dxdx22 isis ψψ22n,mn,m(x(x11,x,x22)dx)dx11dxdx22, which is just a product of separate , which is just a product of separate

probabilitiesprobabilities ψψ22nn(x(x11)dx)dx11 and and ψψ22

mm(x(x22)dx)dx22. However, even though we . However, even though we

label the particleslabel the particles 11 and and 22, we can not distinguish which is in, we can not distinguish which is in dxdx11

and which is inand which is in dxdx22 if they are identical. The mathematical if they are identical. The mathematical

description of identical particles must be the same if we description of identical particles must be the same if we interchange the labels. Therefore, the probability density interchange the labels. Therefore, the probability density

ψψ22(x(x22,x,x11) = ) = ψψ22(x(x11,x,x22))

This equation is satisfied ifThis equation is satisfied if ψψ is either symmetric or antisymmetric: is either symmetric or antisymmetric:

ψψ22(x(x22,x,x11) = ) = ψψ22(x(x11,x,x22),), symmetricsymmetric

oror

ψψ22(x(x22,x,x11) = -) = -ψψ22(x(x11,x,x22),), antisymmetric antisymmetric

The SchrThe Schrödinger Equation for Two Identical Particlesödinger Equation for Two Identical ParticlesFor example, the symmetric and For example, the symmetric and

antisymmetric wave function for the first exited antisymmetric wave function for the first exited state of two identical particles in a infinity square state of two identical particles in a infinity square well:well:

andand

L

x

L

x

L

x

L

xAS

1221 2sinsin

2sinsin

L

x

L

x

L

x

L

xAA

1221 2sinsin

2sinsin


There is an important difference between There is an important difference between antisymmetric and symmetric wave functions. Ifantisymmetric and symmetric wave functions. If n=mn=m, the antisymmetric wave function is identically , the antisymmetric wave function is identically zero for all values ofzero for all values of xx11 and and xx22 , whereas the , whereas the

symmetric function is not. Thus, the quantum symmetric function is not. Thus, the quantum numbersnumbers nn andand mm can not be the same for can not be the same for antisymmetric function. antisymmetric function.

Pauli exclusion principle:Pauli exclusion principle:

No two electrons in an atom can have same No two electrons in an atom can have same quantum numbers.quantum numbers.

The SchrThe Schrödinger Equation in Spherical Coordinatesödinger Equation in Spherical Coordinates

In quantum theory, the electron is described In quantum theory, the electron is described by its wave functionby its wave function ψψ. . The probability of finding The probability of finding the electron in some volumethe electron in some volume dVdV of space is equals of space is equals the product of absolute square of the electron the product of absolute square of the electron wave functionwave function ||ψψ||22 andand dVdV. .

Boundary conditions on the wave function Boundary conditions on the wave function lead to quantization of the wavelengths and lead to quantization of the wavelengths and frequencies and thereby to the quantization of the frequencies and thereby to the quantization of the electron energy. electron energy.


Consider a single electron of mass Consider a single electron of mass mm moving in moving in three dimensions in a region in which the potential three dimensions in a region in which the potential energy is energy is VV. The time independent . The time independent SchrSchrödinger ödinger Equation for such a particle:Equation for such a particle:

For a single isolated atom, the potential energy For a single isolated atom, the potential energy VV depends only on the radial distance depends only on the radial distance r = √xr = √x22 + y + y22 + + zz22 . The problem is then most conveniently treated . The problem is then most conveniently treated using the using the spherical coordinatesspherical coordinates. .

EV

dz

d

dy

d

dx

d

m 2

2

2

2

2

22

2


We will use coordinates We will use coordinates rr, , θθ, and , and φφ,, which related to the which related to the rectangular coordinates rectangular coordinates xx, , yy,, and and zz byby

z = r cosz = r cosθθ, , x = r sinx = r sinθθcoscosφφ, , andand y = y = r sinr sinθθsinsinφφ


(1)

The transformation of the wave term in the equation:The transformation of the wave term in the equation:

Substitution in equation (1) gives (2):Substitution in equation (1) gives (2):

EU

dz

d

dy

d

dx

d

m 2

2

2

2

2

22

2

2

2

2222

22

2

2

2

2

2

sin

11sin

sin

111

d

d

rd

d

d

d

rdr

dr

dr

d

rdz

d

dy

d

dx

d

ErUd

d

d

d

d

d

mrdr

dr

dr

d

mr)(

sin

1sin

sin

1

22 2

2

22

22

2

2

2


The first step in solving this partial differential The first step in solving this partial differential equation is to separate the variables by writing the equation is to separate the variables by writing the wave function wave function ψψ(r,(r,θθ,,φφ)) as a product of functions of as a product of functions of each single variable:each single variable:

ψψ(r,(r,θθ,,φφ) = R(r) f() = R(r) f(θθ) g() g(φφ),),where where RR represent only the radial coordinate represent only the radial coordinate rr; ; ff depends only of depends only of θθ, , andand gg depends only ofdepends only of φφ. . When When this form of this form of ψψ(r,(r,θθ,,φφ)) is substituted into equation (2) the is substituted into equation (2) the partial differential equation can be transformed into partial differential equation can be transformed into three ordinary differential equations, one for three ordinary differential equations, one for R(r)R(r), one , one for for f(f(θθ)) and one forand one for g(g(φφ))..

The potential energy The potential energy U(r)U(r) appears only in equation for appears only in equation for R(r),R(r), which is called the radial which is called the radial equationequation


In three dimensions, the requirement that the wave In three dimensions, the requirement that the wave function be continuous and normalizable introduces three function be continuous and normalizable introduces three quantum numbers, one associated with each spatial dimension. quantum numbers, one associated with each spatial dimension. In spherical coordinates the quantum number associated withIn spherical coordinates the quantum number associated with r r is labeled is labeled nn, that associated with , that associated with θθ is labeled is labeled ℓℓ,, and that and that associated with associated with φφ is labeled is labeled mmℓℓ..

For the rectangular coordinates For the rectangular coordinates x,yx,y,, and and zz the the corresponded quantum numbers corresponded quantum numbers nn11, , nn22,, and and nn33 for a particle in a for a particle in a

three-dimensional square well were independent of one other, three-dimensional square well were independent of one other, but the quantum numbers associated with wave function for but the quantum numbers associated with wave function for spherical coordinates are interdependent. spherical coordinates are interdependent.

Summary of the Quantum NumbersSummary of the Quantum NumbersThe possible values of this quantum numbers are:The possible values of this quantum numbers are:

n = 1,2,3,…..n = 1,2,3,…..

ℓ ℓ = 0,1,2,3,…,(n-1)= 0,1,2,3,…,(n-1)

mmℓℓ = -ℓ, (-ℓ +1),……,-2,-1,0,1,2,….,(ℓ +1),ℓ=0, ±1, ±2,…,±ℓ = -ℓ, (-ℓ +1),……,-2,-1,0,1,2,….,(ℓ +1),ℓ=0, ±1, ±2,…,±ℓ

The number The number nn is called the is called the principal quantum numberprincipal quantum number. It is . It is associated with the dependence of the wave function on the associated with the dependence of the wave function on the distancedistance r r and therefore with the probability of finding the and therefore with the probability of finding the electron at various distances from the nucleus.electron at various distances from the nucleus.

The quantum numbersThe quantum numbers ℓℓ and and mmℓℓ are associated with the are associated with the

angular momentum of the electron and with the angular angular momentum of the electron and with the angular dependence of the electron wave function.dependence of the electron wave function.

The quantum number The quantum number ℓℓ is called the is called the orbital quantum numberorbital quantum number. . The magnitude The magnitude LL of the orbital angular momentum is related to of the orbital angular momentum is related to ℓℓ byby

││L│ = √ ℓ (ℓ +1) ħL│ = √ ℓ (ℓ +1) ħ

Summary of the Quantum NumbersSummary of the Quantum Numbers

The quantum number The quantum number mmℓℓ is called the is called the magnetic quantum magnetic quantum numbernumber, it is relate to the , it is relate to the zz--component of angular momentum. component of angular momentum. Since there is not preferred direction for theSince there is not preferred direction for the zz--axis for any axis for any central force, all spatial directions are equivalent for an isolated central force, all spatial directions are equivalent for an isolated atom. atom.

However, if we will place the atom in an external However, if we will place the atom in an external magnetic field the direction of the field will be separated out magnetic field the direction of the field will be separated out from the other directions. If from the other directions. If zz--direction is chosen for the direction is chosen for the magnetic field direction, than magnetic field direction, than z-z-component of the angular component of the angular momentum of the electron is given by the quantum condition:momentum of the electron is given by the quantum condition:

LLZZ= m= mℓℓħħ

This quantum condition arises from the boundary This quantum condition arises from the boundary conditions on the azimuth coordinate conditions on the azimuth coordinate φφ that the probability of that the probability of finding the electron at some angle finding the electron at some angle φφ11 must be the same as that must be the same as that of finding the electron at angle of finding the electron at angle φφ11+2+2ππ because these are the because these are the same points in the space. same points in the space.


If we measure the If we measure the angular momentum of the angular momentum of the electron in units of electron in units of ħħ,, we we see that the angular-see that the angular-momentum magnitude is momentum magnitude is quantized to the value quantized to the value

√ ℓ √ ℓ (ℓ+1)(ℓ+1) units and that its units and that its component along any component along any direction can have only direction can have only the the 2ℓ + 12ℓ + 1 values ranging values ranging from from -ℓ-ℓ to to +ℓ+ℓ units. On units. On the figure we can see the the figure we can see the possible orientation of possible orientation of angular momentum angular momentum vector for vector for ℓ=2ℓ=2..

Vector-model diagram illustrating Vector-model diagram illustrating the possible values of the possible values of zz-component -component of the angular momentum vector for of the angular momentum vector for the case the case ℓ=2ℓ=2.. The magnitude of The magnitude of L=ħ√6L=ħ√6..

The direction of the angular momentum:The direction of the angular momentum:

If the angular momentum is characterized by the If the angular momentum is characterized by the quantum number quantum number ℓ = 2ℓ = 2,, what are the possible values of what are the possible values of LLZZ,, and what is the smallest possible angle between and what is the smallest possible angle between LL

and the and the zz axis?axis?


That is, That is, nn can be any positive integer;can be any positive integer; ℓℓ can be zero or can be zero or any positive integer up to any positive integer up to (n-1)(n-1);; and and mmℓℓ can have can have (2ℓ+1)(2ℓ+1)

positive values, ranging from positive values, ranging from -ℓ -ℓ to to ++ℓℓ in integral steps. in integral steps. In order to explain the fine structure and to clear up In order to explain the fine structure and to clear up

some difficulties with explanation the table of elements some difficulties with explanation the table of elements Pauli Pauli suggested that in addition to the quantum numbers suggested that in addition to the quantum numbers nn, , ℓℓ,, and and mmℓℓ the electron should have a the electron should have a fourthfourth quantum number quantum number, which , which

could take on just two values. could take on just two values. This fourth quantum number is the This fourth quantum number is the zz--component,component, mmzz, of , of

an intrinsic angular momentum of the electron, called an intrinsic angular momentum of the electron, called spinspin. . The The spinspin vector vector SS relate to this relate to this fourth quantum numberfourth quantum number ss by by │S│ = √s(s+1) ħ│S│ = √s(s+1) ħ and can take only two valuesand can take only two values ±½±½..

Quantum Theory of the Hydrogen AtomQuantum Theory of the Hydrogen Atom

We can treat the simplest hydrogen atom as a We can treat the simplest hydrogen atom as a stationary nucleus, a proton, that has a single moving particle, stationary nucleus, a proton, that has a single moving particle, an electron, with kinetic energy an electron, with kinetic energy pp22/2m/2m.. The potential energy The potential energy U(r)U(r) due to the electrostatic attraction between the electron due to the electrostatic attraction between the electron and the proton isand the proton is

In the lowest energy state, which is the ground state, In the lowest energy state, which is the ground state, the principal quantum numberthe principal quantum number n=1n=1, , ℓ=0ℓ=0,, and and mmℓℓ=0=0..

The allowed energiesThe allowed energies

r

kZerU

2

)(

eVemk

E

nn

EZ

n

emkZEn

6.132

,....3,2,12

2

42

0

202

22

422

Potential energy of an electron in a hydrogen atom. If the Potential energy of an electron in a hydrogen atom. If the total energy is greater than zero, as total energy is greater than zero, as E’E’, the electron is not , the electron is not bound and the energy is not quantized. If the total energy is bound and the energy is not quantized. If the total energy is less than zero, asless than zero, as E E, the electron is bound, than, as in one-, the electron is bound, than, as in one-dimensional problems, only certain discrete values of the dimensional problems, only certain discrete values of the total energy lead to well-behaved wave function. total energy lead to well-behaved wave function.

Energy-level diagram for Energy-level diagram for hydrogen. The diagonal hydrogen. The diagonal lines show transitions lines show transitions that involve emission or that involve emission or absorption of radiation absorption of radiation that obey the selection that obey the selection rules rules ΔℓΔℓ = ±1 = ±1, , mmℓℓ=0=0 or or

±1±1. States with the same . States with the same value of value of nn but with but with different values ofdifferent values of ℓℓ have have the same energy the same energy ––EE00/n/n22,,

where where EE00=13.6 eV=13.6 eV..

The wavelength of the The wavelength of the light emitted by the atom light emitted by the atom relate to the energy relate to the energy levelslevels by by

hf = hc =Ehf = hc =Eii-E-Eff

Energy-level diagram for Energy-level diagram for the hydrogen atom, the hydrogen atom, showing transitions showing transitions obeying the selection rule obeying the selection rule ΔΔl = ±1l = ±1. States with the . States with the same same nn value but value but different different ll value have the value have the same energy, same energy, -E-E11/n/n22, , where where EE11=13.6 eV=13.6 eV, as in , as in the Bohr theory. The the Bohr theory. The wavelength of the Lyman wavelength of the Lyman αα(n = 2 → n =1)(n = 2 → n =1) and and Balmer Balmer αα(n = 3 → n = 2)(n = 3 → n = 2) lines are shown in nm. lines are shown in nm. Note that the latter has Note that the latter has two possible transactions two possible transactions due to the due to the ll degeneracy. degeneracy.

Wave Function and the probability densityWave Function and the probability density

The ground stateThe ground state: In the lowest energy state, the ground : In the lowest energy state, the ground state of the hydrogen, state of the hydrogen, n=1n=1, , ℓ=0ℓ=0,, and and mmℓℓ=0=0, , EE00=13.6eV=13.6eV,, and and

the angular momentum is the angular momentum is zerozero. The wave function for the . The wave function for the ground state isground state is

is the Bohr radius and is the Bohr radius and CC1,0,01,0,0 is a constant that is determined is a constant that is determined

by the normalization. In three dimensions, the normalization by the normalization. In three dimensions, the normalization condition iscondition is

∫│∫│ψ│ψ│22dV = 1dV = 1

where where dVdV is a volume element and the integration is is a volume element and the integration is performed over all space.performed over all space.

00,0,10,0,1

a

Zr

eC

nmmke

a 0529.02

2

0

where

Volume Element in Spherical Coordinates

Volume element in spherical coordinatesVolume element in spherical coordinates

In spherical coordinates the volume element is In spherical coordinates the volume element is

dV = (r sindV = (r sinθθddφφ)(rd)(rdθθ)dr = r)dr = r2 2 sinsinθθddθθddφφdrdr

We integrate overWe integrate over φφ, from , from φφ=0=0 to to φφ=2=2ππ; over ; over θθ, from , from θθ=0=0 to to θθ==ππ; and over ; and over rr from from r=0r=0 to to r=∞.r=∞. The normalization conditions The normalization conditions is thusis thus

Since there is noSince there is no θθ or or φφ dependence in dependence in ΨΨ1,0,01,0,0 the triple integral the triple integral

can be factored in a product of three integralscan be factored in a product of three integrals

1sin

sin

0 0

2

0

2

2

20,0,1

0 0

2

0

222

0

drddreC

drddrdV

a

Zr

This givesThis gives

The remaining integral is of the form The remaining integral is of the form

with with nn a positive integer and with a positive integer and with a>0a>0. .

122

sin

0

2

220,0,1

0

2

2

20,0,1

0

2

0

2

0

0

drerC

drreCdddV

a

Zr

a

Zr

0

dxex axn

This integral can be looked up in a table of integrals This integral can be looked up in a table of integrals

soso

ThanThan

10

!

n

axn

a

ndxex

0

3

30

2

2

40

Z

adrer a

Zr

14

43

302

0,0,1

Z

aC

23

00,0,1

1

a

ZC

soso

The normalized ground-state wave function is thus The normalized ground-state wave function is thus

The probability of finding the electron in a volume The probability of finding the electron in a volume dV dV isis|ψ||ψ|22dVdV

0

23

00,0,1

1 a

Zr

ea

Z

Probability DensitiesProbability Densities

Computer generated probability density Computer generated probability density |ψ||ψ|22 for the ground state for the ground state of the hydrogen. The quantity of the hydrogen. The quantity -e |ψ|-e |ψ|22 can be though of as the can be though of as the electron charge density in the atom. The density is spherically electron charge density in the atom. The density is spherically symmetric (it depends only on symmetric (it depends only on r r and independent of and independent of θθ or or φφ), is ), is greatest at the origin , and decrease exponentially withgreatest at the origin , and decrease exponentially with r r. .


We are more often interested in the probability of finding We are more often interested in the probability of finding the electron at some radial distance the electron at some radial distance rr between between rr and and r+drr+dr. This . This radial probability radial probability P(r)drP(r)dr is is |ψ||ψ|22dVdV, where , where dVdV is the volume of the is the volume of the spherical shell of thickness spherical shell of thickness drdr, which is , which is dV=4πrdV=4πr22drdr. .

The probability of finding the electron in the range from The probability of finding the electron in the range from rr to to

r+drr+dr is thus is thus

and the and the radial probability densityradial probability density is is

drrdrrP 224)(

224)( rrP


For the hydrogen atom in the ground state, the For the hydrogen atom in the ground state, the radial probability density is radial probability density is

00

2

2

3

0

2

20,0,1

22 444)( a

Zr

a

Zr

era

ZerCrrP

Radial Probability DensityRadial Probability Density

Radial probability density Radial probability density P(r)P(r) versus versus r / ar / a00 for the for the

ground state of the hydrogen ground state of the hydrogen atom. atom. P(r)P(r) is proportional to is proportional to rr22ΨΨ22. The value of . The value of rr for which for which P(r)P(r) is maximum is the most is maximum is the most probable distance probable distance r=ar=a00, ,

which is the first Bohr radius.which is the first Bohr radius.

Radial probability density Radial probability density P(r)P(r) vs. vs. r/a r/a00 for the for the n=2n=2 states states in hydrogen. in hydrogen. P(r)P(r) for for l=1l=1 has a maximum at the Bohr has a maximum at the Bohr value value 2222aa00. For . For l = 0l = 0 there is a maximum near this value there is a maximum near this value and a smaller submaximum near the origin. The markers and a smaller submaximum near the origin. The markers on the on the r/ar/a00 axis denote the values ofaxis denote the values of (r/a(r/a00).).

P(r)P(r) vs. vs. r/ar/a00 for the for the n = 3n = 3 state in hydrogen. state in hydrogen.

Probability densityProbability density ΨΨ**ΨΨ for the for the n=2n=2 states in hydrogen. The states in hydrogen. The probability is spherically symmetric for probability is spherically symmetric for l=0l=0. It is proportional to . It is proportional to coscos22θθ for for l=1, m=0l=1, m=0, and to, and to sin sin22 for for l=1, m=±1l=1, m=±1. The probability . The probability densities have rotational symmetry about the densities have rotational symmetry about the zz axis. Thus, the axis. Thus, the three-dimensional charge density for three-dimensional charge density for l=1, m=0l=1, m=0 state is shaped state is shaped roughly like a dumbbell, while that for the roughly like a dumbbell, while that for the l=1l=1, , m=±1 m=±1 states states resembles a doughnut, or toroid. The shapes of these resembles a doughnut, or toroid. The shapes of these distributions are typical for all atoms in distributions are typical for all atoms in SS states ( states (l=0)l=0) and and P P states states (l=1)(l=1) and play an important role in molecular bounding. and play an important role in molecular bounding.

A particle moving in a A particle moving in a circle has angular circle has angular momentum momentum LL. If the . If the particle have a positive particle have a positive charge, the magnetic charge, the magnetic moment due to the moment due to the current is parallel to current is parallel to LL..

Bar-magnet model of Bar-magnet model of magnetic moment. magnetic moment. (a) In an external (a) In an external magnetic field, the magnetic field, the moment experiences a moment experiences a torque which tends to torque which tends to align it with the field. If align it with the field. If the magnet is spinning the magnet is spinning (b), the torque caused (b), the torque caused the system to precess the system to precess around the external around the external field. field.

Example:Example: Probability that electron is in a thin Probability that electron is in a thin spherical shellspherical shell

Find the probability of finding the electron in a thin Find the probability of finding the electron in a thin spherical shell of radius spherical shell of radius rr and thickness and thickness Δr=0.06aΔr=0.06a00

at (a) at (a) r=ar=a00 and (b) and (b) r=2ar=2a00 for the ground state of the for the ground state of the

hydrogen atom.hydrogen atom.

The spin-orbit effect and fine structureThe spin-orbit effect and fine structure

The total angular momentum of an electron in The total angular momentum of an electron in an atom is a combination of the orbital angular an atom is a combination of the orbital angular momentum and spin angular momentum. It is momentum and spin angular momentum. It is characterized by the quantum numbercharacterized by the quantum number jj, which can , which can be either be either │ℓ - ½││ℓ - ½│ or or │ℓ + ½│.│ℓ + ½│. Because of Because of interaction of the orbital and spin magnetic interaction of the orbital and spin magnetic moments, the state moments, the state j = │ℓ - ½│j = │ℓ - ½│ has lower energy has lower energy than the state than the state j = │ℓ + ½│,j = │ℓ + ½│, for for ℓ ≥1ℓ ≥1. This small . This small splitting of the energy states gives rise to a small splitting of the energy states gives rise to a small splitting of the spectral lines called fine structure. splitting of the spectral lines called fine structure.

The Table of ElementsThe Table of Elements

We can treat the simplest hydrogen atom as a We can treat the simplest hydrogen atom as a stationary nucleus, a proton, that has a single moving particle, stationary nucleus, a proton, that has a single moving particle, an electron, with kinetic energy an electron, with kinetic energy pp22/2m/2m.. The potential energy The potential energy U(r)U(r) due to the electrostatic attraction between the electron due to the electrostatic attraction between the electron and the proton isand the proton is

In the lowest energy state, which is the ground state, In the lowest energy state, which is the ground state, the principal quantum numberthe principal quantum number n=1n=1, , ℓ=0ℓ=0,, and and mmℓℓ=0=0..

The allowed energiesThe allowed energies

r

kZerU

2

)(

eVemk

E

nn

EZ

n

emkEn

6.132

,....3,2,12

2

42

0

202

22

42


For atoms with more than one electron, the For atoms with more than one electron, the SchrSchröödinger dinger equation cannot be solved exactly. equation cannot be solved exactly. However, the approximation methods allow to However, the approximation methods allow to determine the energy levels of the atoms and determine the energy levels of the atoms and wave functions of the electrons with high wave functions of the electrons with high accuracy. accuracy.

As a first approximation, the As a first approximation, the ZZ electrons in electrons in an atom are assumed to be noninteracting. The an atom are assumed to be noninteracting. The SchrSchröödingerdinger equation can then be solved, and equation can then be solved, and the resulting wave function used to calculate the the resulting wave function used to calculate the interaction of the electrons. interaction of the electrons.

The state of each electron in an atom is described by The state of each electron in an atom is described by four quantum numbers four quantum numbers n,l,m,n,l,m, and and mmss. .

Beginning with hydrogen, each larger neutral atom adds Beginning with hydrogen, each larger neutral atom adds one electron. The electrons go into those states that will give one electron. The electrons go into those states that will give the lowest energy consistent with the the lowest energy consistent with the PauliPauli exclusion principle: exclusion principle:

No two electrons in an atom can have the same set of No two electrons in an atom can have the same set of values for the quantum numbers n, ℓ, m, and mvalues for the quantum numbers n, ℓ, m, and mss

The energy of the electron is determined mainly by the The energy of the electron is determined mainly by the principal quantum number principal quantum number nn, which is relate to the radial , which is relate to the radial dependence of the wave function, and by the orbital angular-dependence of the wave function, and by the orbital angular-momentum quantum number momentum quantum number ℓℓ..

The dependence of the energy on The dependence of the energy on ℓ ℓ is dueis due to the to the interaction of the electrons in the atoms with each other.interaction of the electrons in the atoms with each other.


The specification of The specification of nn and and ℓ ℓ for each electron in an atom for each electron in an atom is called the electron configuration. is called the electron configuration.

The The ℓℓ values are specified by a code:values are specified by a code: s p d f g hs p d f g h

ℓℓ valuesvalues 0 1 2 3 4 50 1 2 3 4 5The The nn values are referred as shells, which are identified values are referred as shells, which are identified

by another letter code: by another letter code: shell K L M N …..shell K L M N …..nn values values 1 2 3 4 ….. 1 2 3 4 …..

Using the exclusion principle and the restriction of the Using the exclusion principle and the restriction of the quantum numbers (quantum numbers (nn is a positive integer, is a positive integer, ℓ ℓ ranged from ranged from

00 to to n-1n-1, , mm changed from changed from -ℓ-ℓ to to ℓ ℓ in integral steps, and in integral steps, and mmss can can be either be either +½ +½ or or -½-½), we can understand much of the structure ), we can understand much of the structure of the periodic table.of the periodic table.


The Periodic TableThe Periodic TableThe energy required to remove the most loosely The energy required to remove the most loosely

electron from an atom in the ground state is called the electron from an atom in the ground state is called the ionization energyionization energy. This energy is the binding energy of . This energy is the binding energy of the last electron placed in the atom. The the last electron placed in the atom. The ionization ionization energyenergy can be found from: can be found from:

Hydrogen (Z = 1)Hydrogen (Z = 1):: n=1n=1, , ℓ = 0, m = 0, mℓ = 0, m = 0, mss = ±½ = ±½ - 1s - 1s

Helium (Z = 2):Helium (Z = 2): two electrons, in the ground state both two electrons, in the ground state both electrons are in theelectrons are in the K K shell shell, , n=1,n=1, ℓ=0, m=0, ℓ=0, m=0, mms1s1=+½, m=+½, ms2s2=-½ =-½ - 1s- 1s22

Lithium (Z=3):Lithium (Z=3): KK shell (n=1) is completely full, one electron shell (n=1) is completely full, one electron on theon the L L-shell – -shell – 2p2p11

)6.13(2

2

02

2

eVn

ZE

n

ZE effeff

n

States of Hydrogen AtomStates of Hydrogen Atom

nn 11 22 33

ℓℓ 00 00 11 00 11 22

mm 00 00 0,0,±1±1 00 0 ,0 ,±1±1 0, 0, ±1, ±2±1, ±2

mmss ± ± ½½

± ½± ½ ± ½± ½ ± ½± ½ ± ½± ½ ± ½± ½

SubSub

shellshell

22 22 66 22 66 1010

TotalTotal

StatesStates

22 88 1818

The Structure of AtomThe Structure of Atom

The electrons in atom that have same The electrons in atom that have same principal quantum number principal quantum number nn form an form an electron shell:electron shell:

totaltotal

KK n=1n=1 22

LL n=2n=2 88

MM n=3n=3 1818

NN n=4n=4 3232

OO n=5n=5 5050

} 2n2n22

Depending from orbital quantum number Depending from orbital quantum number ℓ ℓ the the electrons forms subshells:electrons forms subshells:

nn ShellShell Total Total elect.elect.

ss((ℓ=0)ℓ=0)

pp

((ℓ =1)ℓ =1)

dd

((ℓ =2)ℓ =2)

ff

((ℓ =3)ℓ =3)

gg

((ℓ =4)ℓ =4)

numbernumber

11 KK 22 -- -- -- -- 22

22 LL 22 66 -- -- -- 88

33 MM 22 66 1010 -- -- 1818

44 NN 22 66 1010 1414 -- 3232

55 OO 22 66 1010 1414 1818 5050

Number of electrons in subshell

Distribution Electrons in atomsDistribution Electrons in atomsZZ ElementElement KK LL MM nnℓℓZZ

1s1s 2s 2p2s 2p 3s3p3d3s3p3d

11 HH 11 - -- - 1s1s

22 HeHe 22 - -- - 1s1s22

33 LiLi 22 1 -1 - 1s1s22, 2s, 2s

44 BeBe 22 2 -2 - 1s1s22, 2s, 2s22

55 BB 22 2 12 1 1s1s22, 2s, 2s22, 2p, 2p

66 CC 22 2 22 2 1s1s22, 2s, 2s22, 2p, 2p22

77 NN 22 2 32 3 1s1s22, 2s, 2s22, 2p, 2p33

88 OO 22 2 42 4 1s1s22, 2s, 2s22, 2p, 2p44

99 FF 22 2 52 5 1s1s22, 2s, 2s22, 2p, 2p55

1010 NeNe 22 2 62 6 1s1s22, 2s, 2s22, 2p, 2p66

For example the structure for oxygen, For example the structure for oxygen, OO, , 1s1s22, 2s, 2s22, 2p, 2p44 – it mean that – it mean that 22 electrons electrons are in the state with are in the state with n=1n=1 and and ℓ=0ℓ=0; ; 22 electrons in the state with electrons in the state with n=2n=2 and and ℓ=0ℓ=0; ; and and 44 electrons in the state with electrons in the state with n=2n=2 and and ℓℓ=1=1. .

1. Effective Nuclear Charge for an Outer 1. Effective Nuclear Charge for an Outer ElectronElectron

Suppose the electron cloud of the outer electron Suppose the electron cloud of the outer electron in the lithium atom in the ground state were completely in the lithium atom in the ground state were completely outside the electron clouds of the two inner electrons, outside the electron clouds of the two inner electrons, the nuclear charge would be shielded by the two inner the nuclear charge would be shielded by the two inner electrons and the effective nuclear charge would be electrons and the effective nuclear charge would be ZZ''e=1ee=1e. Then the energy of the outer electron would . Then the energy of the outer electron would be be –(13.6eV)/2–(13.6eV)/222=-3.4eV=-3.4eV. However, the ionization . However, the ionization energy of lithium is energy of lithium is 5.39eV5.39eV, not , not 3.4eV3.4eV. Use this fact to . Use this fact to calculate the effective nuclear charge calculate the effective nuclear charge ZZeffeff seen by the seen by the

outer electron in lithium. outer electron in lithium.

2. The Effective Charge of the Rb Ion2. The Effective Charge of the Rb Ion

The The 5s5s electron in electron in rubidiumrubidium sees an effective sees an effective charge of charge of 2.771e2.771e. Calculate the ionization . Calculate the ionization energy of this electron.energy of this electron.

3. Determining Z3. Determining Zeffeff experimentally experimentally

The measured energy of a The measured energy of a 3s3s state of sodium is state of sodium is -5.138eV-5.138eV. Calculate the value of . Calculate the value of ZZeffeff..

ExampleExample

The double charged ion The double charged ion NN+2+2 is formed by is formed by removing two electrons from a nitrogen atom. (a) What removing two electrons from a nitrogen atom. (a) What is the ground state electron configuration for the is the ground state electron configuration for the NN+2+2 ion? (b) Estimate the energy of the least strongly ion? (b) Estimate the energy of the least strongly bond level in the bond level in the LL shell of shell of NN+2+2. .

The double charged ion The double charged ion PP+2+2 is formed by is formed by removing two electrons from a phosphorus atom. (c) removing two electrons from a phosphorus atom. (c) What is the ground-state electron configuration for the What is the ground-state electron configuration for the PP+2+2 ion? (d) estimate the energy of the least strongly ion? (d) estimate the energy of the least strongly bound level in the bound level in the MM shell of shell of PP+2+2. .

Electron Interaction Energy in HeliumElectron Interaction Energy in Helium

The ionization energy for helium is The ionization energy for helium is 24.6 eV24.6 eV. .

(a) Use this value to calculate the energy of (a) Use this value to calculate the energy of interaction of the two electrons in the ground state interaction of the two electrons in the ground state of the helium atom.of the helium atom.

(b) Use your result to estimate the average (b) Use your result to estimate the average separation of the two electrons. separation of the two electrons.

Angular Momentum of the Exited Level Of HydrogenAngular Momentum of the Exited Level Of Hydrogen

Consider the Consider the n=4n=4 state of hydrogen. (a) What is state of hydrogen. (a) What is the maximum magnitude the maximum magnitude LL of the orbital angular of the orbital angular momentum? (b) What is the maximum value of momentum? (b) What is the maximum value of LLZZ? (c) What is the minimum angle between and ? (c) What is the minimum angle between and

Z-axis? Give your answers to (a) and (b) in terms Z-axis? Give your answers to (a) and (b) in terms of .of .

L

A Hydrogen Wave Function.A Hydrogen Wave Function.

The ground–state wave function for the hydrogen (The ground–state wave function for the hydrogen (1s 1s state) is:state) is:

(a) Verify that this function is normalized. (b) What is (a) Verify that this function is normalized. (b) What is the probability that the electron will be found at a the probability that the electron will be found at a distance less than distance less than aa from the nucleus?from the nucleus?

a

r

s ea

r

31

1)(

Atomic SpectraAtomic Spectra

Atomic spectra include optical spectra and Atomic spectra include optical spectra and X-X-rayray spectra. Optical spectra result from spectra. Optical spectra result from transmissions between energy levels of a single transmissions between energy levels of a single outer electron moving outer electron moving in the field of the nucleus in the field of the nucleus and core electrons of the atom. and core electrons of the atom.

Characteristic Characteristic X-rayX-ray spectra result from the spectra result from the excitation of a inner core electron and the excitation of a inner core electron and the subsequent filling of the vacancy by other electrons subsequent filling of the vacancy by other electrons in the atom.in the atom.

Selection Rules

Transition between energy states with the Transition between energy states with the emission of a photon are governed by the emission of a photon are governed by the following selection rulesfollowing selection rules

ΔΔmmℓℓ = 0 or ±1 = 0 or ±1

ΔℓΔℓ = ±1 = ±1

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Atomic Physics