Ch40 Young Freedman1

8/12/2019 Ch40 Young Freedman1

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Chapter 40/41: Quantum Mechanics

Wave Functions & 1DSchrodinger Eq

Particle in a Box

Wave function

Energy levels

Potential Wells/Barriers &Tunneling

The Harmonic Oscillator The H-atom


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Wave Equation for a String

For a wave on a string (1D) moving with speed , any wave functionmust satisfy the wave equation (Ch. 15):

,y x t

2 2

2 2 2

, ,1y x t y x t

x v t

,y x t

v

It has the following sinusoidal wave function as its fundamental solution:

, cos siny x t A kx t B kx t

where is the wave number and is the angular frequencyof the wave. [A andB determines the amplitude and phase of the wave.]

2k 2 f


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By substituting the fundamental wave function into the PDE, we arrived atthe algebraic relation (dispersion relation) that and must satisfy:


22

2k

v

,

cos sin sin cosk

y x tA kx t B kx t A kx t B x

xk k t

x

or vk

k

check...

1. Each spatial derivative of will pull out one k: ,y x t

So, the 2ndorder spatial derivative gives,

2

2

2

2, cos siny x t

A kx t x tk B k

x

k

(Obviously, dontforget the signs.)


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,cos sin sin cos

y x tA kx t B kx t A kx t B kx t

t t

check...

2. Each time derivative of will pull out one : ,y x t

So, the 2ndorder time derivative gives,

2

2

2

2, cos siny x t

A kx t B kx tt

(Again, dont forgetthe signs.)

Putting these back into the wave equation, we then have,

2 22 2 2

, ,1y x t y x t

x v t

2

2 2

2 2

cos sin

1cos sin

k kA kx t B kx t

A kx t B kx t

v

22

2k

v


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Thus, the fundamental property of a mechanical wave


Putting the definitions for and back into the Dispersion relation, wehave the familiar relation for wavelength, frequency, and wave speed.

2 22 2 2

, ,1y x t y x t

x v t

v f22

f v

k

22

2orv f k

v

is intimately linked to the form of the wave equation !

Now, we will now try to use the same argument to find a wave equation for aquantum wave function.


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Now, from the de Broglie relations, the energy and momentum of this

quantum free particle must be related to its wave number and angularfrequency through:

Since the reference point for is arbitrary, we can simply take .

Then, the energyEof a free particle will simply be its kinetic energy,

Wave Equation for a Quantum FreeParticle

A free particle has no force acting on it. Equivalently, the potential energy

( )U x

k

0xF dU x dx ( )U x

( ) 0U x

2 2 221

2 2 2

m v pE mv

m m

must be a constant for allx, i.e.,

22

hE hf f

22h hp k

2 2

2

k

m

2

2

pE

m

(non-relativistic)


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We now assume the same fundamental sinusoidal form for the wave function ofa quantum free particle with mass m, momentum and energy :


Thus, a correct quantum wave function for this free particle must satisfy thisquantum dispersion relation for and :k

E

2 2

*2

k

m

, cos sinx t A kx t B kx t

p k

(non-relativistic)

Recall from our discussion on the mechanical wave, we have the following:

t

x

take out an over all kfactor from ,x t

take out an over allfactor from ,x t


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2 2

2

k

m

We can deduce that the PDE for the quantum wave function for this free particle

must involves:


So, from the quantum dispersion relation,

Putting in the other constants and one additional fitting constant C, we have,

t

2

2x

222

, ,

2

x t x tC

m x t


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Now, we substitute into our quantum wave function

into the proposed wave equation to solve for the fitting constant:

, cos sinx t A kx t B kx t

22 2

2 2

2, cos sin

2 2x t Ak kx t Bk kx t

m x m

2 2

cos sin2

kA kx t B kx t

m

,sin cos

x tC C A kx t B kx t

t

cos sinCB kx t CA kx t


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Equating the two terms and using the equality , we have,2 2

2

k

m

cos sinA kx t B kx t cos sinCB kx t CA kx t

A CB

B CA

In order for this equality to be true for all , all coeffs for cos and sinmust equal to each other,

,x t

Substituting the first eq into the second, we have,

2 1B C CB C

Thus, the fitting constant is where .

2 2

2

k

m

1i C i


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With , the free particle quantum wave function can also bewritten in a compact exponential form using the Eulers formula,


Then, finally, putting everything together, we have the desired wave equationfor a quantum free particle,

B CA iA

222

, ,

2

t x ti

m x t

, cos sinx t A kx t i kx t

, i kx t x t Ae

This is the 1D Schrodingers Equation for a free particle.

(quantum wave function for a free particle)


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Free Particle Wave Function &Uncertainty Principle

The wave function for a free particle isa complex function with sinusoidal real

and imaginary parts

Afreeparticle exists in all space , ,

& (wave function extends into all space & time)

but 0 & 0 (energy and momentum is fixed)

x t

p E

Note: 2x p can still be satisfied.2t E &


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More Realistic Particle (Wave Packets)

Under more practical circumstance, a particlewill have a relatively well defined positionand momentum so that bothx and p willbefinite with limited spatial extents.

A more localizedquantum particle can notbe apure sine wave and it can be described by awave packet with a combination of many sinewaves.

( , ) ( ) i kx t x t A k e dk

(a linear combinationof many sine waves.)

The coefficientA(k) gives the relative proportion of

the various sine waves with diff. k (wave number).


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Wave Packets

Recall: Combination of two sine waves more localized than a pure sine wave.


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Wave Packets (characteristic)

p smaller xbigger


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Wave Packets (characteristic)

pbigger x smaller

The is consistent with: !x p


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Quantum Wave Function

In QM, the matter wavepostulated by de Broglie is described by a complex-valuedwavefunction(x,t) which is the fundamental descriptor for a quantumparticle.

x,t

Re/Im (x,t)

(x,t) is a complex-valued function ofspace and time.

1. Its absolute value squaredgives the probability of finding theparticle in an infinitesimal volume dxat time t.

2. For any Q problem: The goal is to find for the

particle for all time. Physical interactions involves

operations (A) on this wavefunction:

Experimental measurements willinvolve the products,

2( , )x t dx

( , )x t

( , )A x t

( , )A x t( , )x t


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Since is a probability, it has to be normalized !

Wave Function and Probability2 *

( , ) ( , ) ( , )x t x t x t

is called the probability distribution function (orprobability density)

In other words, 2

xdx

is theprobability in finding the particle inthe interval at time t.[ , ]x x dx

(shaded area)

2( ) ( , )p x dx x t dx

2( ) ( , ) 1p x dx x t dx

(At any instance of time t, the particle must be somewhere in space !)

2( , )x t dx

(Similar to the intensity of an electric field:being proportional to the # of photons.)

2I E


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The Schrodinger EquationIn Classical Mechanics, we have the Newtons equation which describes the

trajectory x(t) of a particle:

mF x

In EM, we have the wave equation for the propagation of theE, B fields:

2 2

2 2 2

, 1 ,E B E B

x c t

(derived from Maxwells eqns)

In QM, Schrodinger equationprescribes the evolution of the wavefunction

for a particle in timet and spacex under the influence of a potential energy U(x),

2 2

2

( , ) ( , )( ) ( , )

2

x t x tU x x t i

m x t

(general 1D Schrdinger equation)

U(x)


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The 1D Schrodinger Equation

As we have see,

- thefirst term (2ndorder spatial derivative term) in the Schrodinger equationis associated with the Kinetic Energy of the particle

- the last term (the 1st order time derivative term) is associated with the total

energy of the particle- together with the Potential Energy term U(x) (x)

the Schrodinger equation is basically a statement on the conservation ofenergy.

KE PE Total E+ =

2 2

2( , ) ( , )( ) ( , )2x t x tU x x t i

m x t


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Stationary StatesFor most problems, we can factor out the time dependenceby assuming the

following harmonic form for the time dependence,

( , ) ( ) i tx t x e

(Recall the free particle case: .)( , ) ikx i t x t Ae e

With , we can rewrite the time exponent in terms ofE,

/( , ) ( ) iEtx t x e

/E

is a state with a definite energyEand is called a stationary state.( , )x t

Note that,2 * * / /( , ) ( , ) ( , ) ( ) ( )iEt iEt x t x t x t x e x e

2* ( / / ) *( ) ( ) ( ) ( ) ( )i Et Et x x e x x x

( )x is called the time-independent wave function.


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The Time-Independent SchrdingerEquationSubstituting this factorization into the general time-dependent Schrodinger

Eq, we have

/ / /( , )

( () )( )iEt iE iE ttx t iE

i i i xx e E xe et t

and,2 2

2 2/( , ) ( ) iEtx t d x e

x dx

RHS

LHS

2/

2

2

( )

2

iEtd xe

dxm

/( )( ) iEtx eU x /( ) iEtE x e

2 2

2

( )( ) ( ) ( )

2

d xU x x E x

m dx

(time dependence canbe cancelled out !)


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More on Wavefunction

In general, the probability in finding the particle in the interval [a,b] is givenby:2

( )b

aba

p x dx

Note: is notthe probability densityis the probability density.

( )x

Other physicalobservables

can be obtained from

(x

) by the following operation:

2( )x

a b x

p(x)

example (positionx):2

( ) ( )x xp x dx x x dx

is called the expectation value (ofx): it is the experimental

value that one should expect to measure in real experiments !

x


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More on Wavefunction

In general, any experimental observable (position, momentum, energy, etc.)O(x) will have an expectation value given by:

2( ) ( )O O x x dx

Expectation values of physically measurable functions arethe only experimentally accessible quantities in QM.

Wavefunction itself is not a physically measureablequantity.

( )x

Note:

O can bex, p, E, etc.


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Standard Procedure in Solving QMProblems with Schrodinger Equation

(with the time-independent Schrodinger Equation)

Given: A particle is moving under the influence of apotential U(x).

Examples: Free particle: U(x) = 0

Particle in a box:

Barrier:

HMO:

0, 0( )

,

x LU x

elsewhere

0 , 0( )0,

U x LU x

elsewhere

21( ) '2

U x k x


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Standard Procedure in Solving QMProblems with Schrodinger Equation

Solve time-independent Schrodinger equation for (x) as a function of energyE,

with the restrictions:

and are continuous everywhere for smooth U(x).

is normalized, i.e.,

Boundedsolution:

( )x( )d x

dx

( )x

2

( ) 1x dx

( ) 0x as x

Then, expectation values of physical measurable quantities can be calculated.


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Particle in a Box

A 1-Dbox with hardwalls:

(0) ( )U U L (non-penetrable)

A free particle inside the box:( ) 0U x (inside box)

No forces acting on the particle

except at hard walls.

P (in x) is conserved betweenbounces

|P| is fixed but P switches

sign between bounces.

Classical Picture


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Particle in a Box (Quantum Picture)The situation can be described by the following potential energy U(x):

0, 0( )

,

x LU x

elsewhere

2 2

2

( )( ) ( ) ( )

2

d xU x x E x

m dx

The time-independentSchrodinger equation is:

Recall, this is basically

KE PE Total E+ =

Problem statement: For a given U(x), what are the possible wave functions (x)and their corresponding energiesE ?


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Inside the box, , U(x) = 0, and the particle is free. From before,

we know that the wave function for a free particle has the following form:

Wave functions for a Particle in a Box0 x L

1 2( ) ikx ikx

inside x A e A e (linear combination of the two

possible solutions.)

whereA1 andA2 are constants that will be determined later.

Outside the box, , and the particle cannot exist outside the box and( )U x

( ) 0outside x (outside the box)

At the boundary,x = 0 andx =L,the wavefunction has to be continuous:

(0, ) (0, ) 0inside outsideL L

2 2

2

kE

m


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Wavefunctions for a Particle in a Box

Let see how this boundary condition imposes restrictions on the two

constants,A1 andA2, for the wave function.

Using the Eulers formula, we can rewrite the interiorwave function in termsof sine and cosine:

1 2

1 2 1 2

( ) cos sin cos sin

cos sin

inside x A kx i kx A kx i kx

A A kx i A A kx

Imposing the boundary condition atx = 0,

1 2 1 2(0) cos 0 sin 0inside A A i A A 1 2 0A A

1 2A A 1

( ) 2 sin sininside

x iA kx C kx (where C=2iA1

)

ikxe ikxe


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Wavefunction for a Particle in a Box

2 2, 1,2,3,

n

n

Ln

k n

/ 2

3 / 2

L must fit an integral number ofhalf-wavelengths: 2

nL n

So, ( ) sin sin ,

1,2,3,

n n

nx C k x C x

L

n

(similar to standing waves on a cramped string)

2

5 / 2


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Quantized Energies for a Particle in aBox

Since the wave number kn is quantized, the energy for the particle in the box is

also quantized:

22 2 2 2 2 2 2 2

2 2or , 1,2,3,

2 2 2 8n

n

k n n n hE n

m m L mL mL

Note: the lowest energyis not zero:

2

1 2 08

h

E mL

n = 0 gives (x) = 0and it means noparticle.

(n is called thequantum number)


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Probability in Finding the Particle

Notes:

The positions for the particles areprobabilistic. We just know that ithas to be in the box but the exactlocation within the box is uncertain.

Notall positions betweenx = 0 andL are equally likely. In CM, allpositions are equally likely for theparticle in the box.

There are positions where theparticle haszeroprobability to befound.


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Probability and Normalization

Although we dont know exactly where the particle might be inside the box,we know that it has to be in the box. This means that,

2

( ) 1x dx

2 2 2

0sin 1

2

L n LC x dx C

L

(normalization condition)

So, the normalization condition fixes the final free constant Cin thewavefunction, . This then gives,2C L

2( ) sinn

n xx

L L

(particle in a box)


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Time Dependence

Note that with (x) found, we can write down the full wavefunction for the

time-dependentSchrodinger equation as:/( , ) ( ) iEt

n nx t x e

Note that the absolute value for is unity, i.e.,

/2( , ) sin iEtn

n xx t e

L L

/iEte

2/ / / 0 1iEt iEt iEt e e e e

so that |n(x,t)|2 = | (x)|2 is independentof time and probability density in

finding the particle in the box is also independentof time.

recall E hf


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Finite Square-Well Potential

Square-well with finite height

0 ,( )0, 0

U elsewhereU x

x L

In Newtons mechanics, a particlewill be trapped inside the well if

E< U0. In QM, such a trappedstate is called a bound state.

IfE > U0, then the particle is not bound.

For the infinitely deep well (as in theparticle in a box problem), all states

are bound states.

For afinite square-well, there willtypically be only a finite number of

bound states.


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Finite Square-Well PotentialSimilar to the particle in a box problem, U(x) = 0 inside the well, we have

22

2

1 2

( ) 2( ), where

( ) ikx ikx

d x mE k x k

dx

x A e A e

andA, B are constants to be determined by boundary conditions and normalization.

(inside the well)

But for a finite square-well potential, the wavefunction is not identically zerooutside the well. The Schrodinger equation is given by:

2

2

20

02 2( )( ) ( ), where

mU E

m U Ed xx x

dx

(outside the well)

( ) cos sininside x A kx B kx or,


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Finite Square-Well Potential

( ) x xx Ce De

Since U0 >E, is positive and the wavefunction outside the well is given in

terms of exponentials instead of harmonic functions:

where CandD are constants to be determined by B.C. and normalization again.

wavefunction must remainfinite (not blowing up) at large |x|

For this problem, there is a new type of B.C. at large distances from the origin:

0 ( ) and ( )x x

x x Lx Ce x De


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Finite Square-Well PotentialAs in the particle in a box problem, both and must also be

continuous atx = 0 andx =L.

( )x ( ) /d x dx

Matching 0 ( ), ( ), and ( )x inside x Lx x x atx = 0 andx =L will enforce a certain

set of allowed functions to be fitted within the well and the bound stateenergy is correspondingly quantized.


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Example: e in a Square-Well/QuantumDotAn electron trapped in a Square-Well potential with widthL = 0.5nm (~size ofan atom)

a) What is ground state energy if thiswell is infinitely deep U0 = instead ?

b) Now, back to a finite well with

,1E

0 ,16 9.0U E eV

22 342 2

,1 22 31 9

19

1.055 10

2 2 9.11 10 0.50 10

2.4 10 1.50

J sE

mL kg m

J eV

The energy levels for the finite well aregiven as shown on the next slide.

(not derived here)


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Example: e in a Square-Well/QuantumDot

= 0.94eV

= 3.6eV

= 7.6eV

What is the wavelength of light released ifthe electron was originally at the 1st

excited state (n=2) and relaxed back to theground state (n=1)?

2 1

2 1

1240460

3.6 0.94

hchf E E

eV shcnm

E E eV eV


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Example: e in a Square-Well/QuantumDot

DQdot

Application: Quantum dots are nanometer-sized

particles of a semi-conductor (such as cadmiumselenide or gallium arsenide).

An electron within a quantum dot behaves much

like a particle in a finite square well potential.

When a quantum dot is illuminated by aultraviolet light, the electron within the quantumdot can be excited to a higher energy state (let say, n=3) from ground state (n=1).When it relaxed back to the ground state thru the intermediate state (n=2): [32and 21] photons with lower energy (longer wavelengths in the visible range)can be observed (fluorescence) !


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Tunneling Through a Barrier

Consider the followingpotential barrier:

0 , 0( )0,

U x LU x

elsewhere

A quantum particle with mass m and energyEis traveling from the left to the right.

Classical Expectation (withE< U0):

In the regionx < 0, the particle isfreebut when it reachesx = 0, the particle willhit a wall since itsEis less than the potential atx =0. It will be reflected backand it could not penetrate the barrier!


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Tunneling Through a BarrierQuantum Picture:

x < 0 andx >L (free space):

The wavefunction for a freeparticle with definiteEand

P is sinusoidal, eikx

or e-ikx

.

(inside the barrier):0 x L

0E U wavefunction is adecaying exponential e-x.

exponential function within barrier


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Tunneling Through a BarrierIf the energy is high enough and the barrier is not too wide so that the

exponential decay does not significantly diminish the amplitude of theincidence wave, then there is a non-zero probability that a quantum particlemight penetrate the barrier.

(reduced amplitude reduced probability but not zeroprobability !)

The transmission probability Tcan be solved from the Schrodinger equationby enforcing the boundary conditions:

020 0

2, , 16 1L m U E E E

T Ge GU U

(forE/U0 small)


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Application of Tunneling (STM)Scanning Tunneling Microscope (STM):

In a STM, an extremely sharp conducting needle isbrought very close to a surface that one wants toimage.

When the needle is at a positive potential with

respect to the surface, electrons from the surface cantunnel through the surface-potential energy barrier.

The tunneling currentdetected will vary

sensitively on theseparationL of thesurface gap and

these variations can be used to map surface features.


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The Harmonic OscillatorClassically, the harmonic oscillator can be envisioned as a mass m acted on by

a conservative force: (Hookes Law: mass on a spring). Its associatedpotential energy is the familiar:

'F k x

21( ) '2

U x k x

For a classical particle with energyE,the particle will oscillate sinusoidally

aboutx = 0 with an amplitudeA andangular frequency .'k m

where k is the spring constant.


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The Harmonic OscillatorThe Harmonic Oscillator is important since it is a good approximation for

ANY potential near the bottom of the well.( )U x


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The Harmonic Oscillator

For the Quantum analysis, we will use the same form of the potential energy

for a quantum Harmonic Oscillator.

2 22

2

( ) 1' ( ) ( )

2 2

d xk x x E x

m dx

And we have the following quantized energies:

22

2 2

( ) 2 1' ( )

2

d x mk x E x

dx

The solutions for this ordinary differential equation with the boundarycondition are called the Hermite functions:

or

( ) 0x as x

Boundary condition consideration: U(x) increases without bound as x so that the wavefunction for particle with a given energyEmust vanish at largex.

2' 2( ) mk xx Ce

1, 0,1,2,

2n

E n n

(ground state n=0)


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The Harmonic OscillatorHermite Functions

note: similar to previousexamples, the lowestE

state is notzero.

note: wavefunction penetration into

classically forbidden regions.

1, 0,1,2,

2n

E n n


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The Harmonic OscillatorProbability Distribution Function:

Classically, the particle with energyEwill slow down as it climbs up on bothside of the potential hills and it will

spend most of its time near .x A Theblue curve depicts this classicalbehavior and the QM ~ CM as thequantum number n increases.


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The H-atomImportant pre-notes: e- does notexist in well-definedcircular orbits around

the nucleus as in the Bohrs model. e- in a H-atom should be envisioned as a cloud or

probability distribution function.

The size and shape of this cloud is described by the wavefunction for the H-atom and it can be explicitly calculated from the Schrodinger equation:

2 2 2 2 2

2 2 2

0

1

2 4

eE

m x y z r

(in 3D)


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The H-atom

In the Schrodinger equation, we haveexplicitly included the Coulomb potentialenergy term under which the electroninteracts with the nucleus at the origin:

2

0

1( ) ,

4

eU r

r

2 2 2

r x y z is the radius in

spherical coordinates.


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Electron Probability DistributionsIn 3D, the probability in finding the electron in a given volume dVis given by,

2( , , )x y z dV

A good way to visualize this 3D probability distribution is to consider athin spherical shell with radius rand thickness dras our choice for dV:

24dV r dr

We denote the probability of finding the electron within this

thin radial shell as the radial probability distributionfunctionP(r) with:

2 2 2( ) 4P r dr dV r dr

r

dr


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Electron Probability DistributionsExamples of the 3-D probability distribution function ||2 (electron cloud):

2110

24 5.29 10a m

me

is the Bohrs radius which wehave seen previously.

The corresponding radial probability distribution functionP(r):

M El t P b bilit


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More Electron ProbabilityDistributions


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Quantum NumberRecall that for a particle in a 1D box, we have one quantum number for the totalenergy of the particle.

It arises from fitting thewavefunction [sin (nx/L)] within abox of lengthL (quantization).

/ 2

3 / 2

2

5 / 2

In the H-atom case, we are in 3D, the fitting of the wavefunction inspace will result in 2 additional quantum numbers (a total of 3).


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Quantum Numbers

1. n Principle Quantum Number: related to the quantization of the main energylevels in the H-atom (as in the Bohrs model).

2

13.6, 1, 2,3,

n

eVE n

n

The other two related to the quantization of the orbital angular momentum ofthe electron. Only certain discrete values of the magnitude and the component

of the orbital angular momentum are permitted:


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Magnetic Quantum Number

Illustrations showing the relation between L andLz.


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Zeeman EffectExperimentally, it was found that under a magnetic field, the energy levels of

the H-atom will splitaccording to the magnetic quantum number ml.

Semi-classical explanation:

e-B

L

e-

orbits around the nucleus and it forms acurrent loop. Lz measures the orientation ofL with respect to B and thus affects theenergy level of the H-atom.


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Anomalous Zeeman Effect

Additional experiments shows that some of the Zeeman lines are further split.

Predicted withalonelm


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Electron SpinsIn 1925, using again semi-classical model, Samuel Goudsmidt and GeorgeUhlenbech demonstrate that this fine structure splitting is due to the spinangular momentum of the electron and this introduces the 4th quantum number.

4. Spin Quantum Number: The electron has another intrinsic physicalcharacteristic akin to spin angular momentum associated with a spinning top.

Pauli and Bohr

This quantum characteristic did not come out fromSchrodingers original theory. Its existence requiresthe consideration of relativistic quantum effects(Diracs Theory).

1,

2z s sS m m

The direction of the spin angular momentum Sz of theelectron is quantized:

1( 1) , 2

sS s s s m


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Pauli Exclusion PrincipleIn order to understand the full electronic structures of the all the elements beyondthe simple single-electron H-atom, we need another quantum idea. In 1925,Wolfgang Pauli proposed the Paulis Exclusion Principle:

no two electrons can occupy the same quantum-mechanicalstate in a given system, i.e., no two electrons in an atom can

have the same set of value for all four quantum numbers (n, l,ml, ms).

The Paulis Exclusion Principle + the set of the four quantum numbers give thecomplete prescription in identifying the ground state configuration of e-s for all

elements in the Periodic Table. Then, all chemical properties for all atomsfollow !

Additional electrons cannot all crowded into the n = 1 state due to thePaulis Exclusion Principle and they must distribute to other higher levels

according to the ordering of the four quantum numbers.


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Wavefunction Labeling SchemeWe have identified 4 separate quantum numbers for the H-atom (n, l, ml, ms).

For a given principal quantum number n, the H-atom has a given energy andthere might be more than one distinct states (with additional choices for theother three quantum numbers). The fact that there are more than one distinctstates for the same energy is call degeneracy.

And, states with different orbitalquantum numbers are labeled as:

0 :

1:2 :

3:

4 :

5 :

l s subshell

l p subshelll d subshell

l f subshell

l g subshell

l h subshell

Historically, states with differentprincipal quantum numbers arelabeled as:

1:2 :

3 :

4 :

n K shell

n L shell

n M shell

n N shell


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Wavefunction Labeling Scheme

ml andms arenot labeled bythis scheme.


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Filling in the Ground State: Example

1 filled, 1free space

H-atom (Z = 1 one e-)

n = 1, l = 0

E

Helium (Z = 2 two e-)

n = 1, l = 0

E the lowestlevel isnow full

Lithium (Z

= 3 threee-

)

n = 1, l = 0

E

Last electron must go to n=2, l=0level by Paulis ExclusionPrinciple.

n = 2, l = 0

n = 2, l = 1n = 2 level

1 0 1l

m


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Filling in the Ground State: Example

Sodium (Z = 11)

n = 1, l = 0

En = 2, l = 0

n = 2, l = 1

n = 3, l = 0

1 0 1lm

Spectroscopic Notation in the Periodic


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Spectroscopic Notation in the PeriodicTableTypically, only the outer most shell (including the subshells within the outer

most shell) is labeled.

1

2

2 4

1

1

2 2

H s

He s

O s p

# of e- in that subshell

shell n value subshell l value

Z = 8outer shellis n =2

two subshells(l =0 and l = 1)

s p

( 0) : 0

( 1) : 1,0,1l

l

s l m

p l m

only 2 max slots

6 max slots with4 taken

8 electrons to fill, 2 will fill Kshelland 6 remaining will need to go toL shell:


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Ground-State Electron Configurations

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Ch40 Young Freedman1