Lecture V

Hydrogen Atom

dr hab. Ewa Popko

Niels Bohr1885 - 1962

Bohr Model of the Atom• Bohr made three assumptions (postulates)• 1. The electrons move only in certain circular orbits, called

STATIONARY STATES. This motion can be described classically

• 2. Radiation only occurs when an electron goes from one allowed state to another of lower energy.

• The radiated frequency is given by hf = Em - En

where Em and En are the energies of the two states• 3. The angular momentum of the electron is restricted to

integer multiples of h/ (2p) =

mevr = n (1)

zyxEzyxzyxVzyxm

,,,,,,2 2

2

2

2

2

22

=

zyxEzyxH ,,,,ˆ =

The Schrödinger equationThe hydrogen atom

The potential energy in

spherical coordinates

(The potential energy function is spherically symmetric.)

Partial differential equation with three independent variables

rerV

2

041)(p

=

The spherical coordinates

(alternative to rectangular coordinates)

For all spherically symmetric potential-energy functions:( the solutions are obtained by a method called separation of variables)

)()(,,,),,( , ll mmlnllmnl rRYrRrzyx ==

Radial function Angular function of and

The hydrogen atom

The functions and are the same for every spherically symmetric potential-energy function.

Thus the partial differential equation with three independent variables

three separate ordinary differential equations

The solutionThe solution is determined by boundary conditions: - R(r) must approach zero at large r (bound state -

electron localized near the nucleus); and must be periodic: (r,, and (r,,2p describe the same point, so

=2p; and must be finite.

Quantum numbers:

n - principal l – orbital ml - magnetic

2220

2

4 132

n

eEn =p

Principal quantum number: n

The energy En is determined by n = 1,2,3,4,5,…;

216.13

neVEn =

E = - 13.6 eV

- 3.4 eV

Ionized atom

n = 1

n = 2

n = 3

reduced mass

Ne

Ne

mmmm

=

)1( = llL ...,2,1,0=l

Quantization of the orbital angular momentum.

The possible values of the magnitude L of the orbital angular momentum L are determined by the requirement, that the function must be finite at =0 and =p.

There are n different possible values of L for the n th energy level!

Orbital quantum number

)1( = llLLz

lmllm ll )1( lml = ...,2,1,0

Quantization of the component of the orbital angular momentum

lz mL =

Quantum numbers: n, l, m

l – orbital quantum numberl - determines permitted values of the orbital angular momentum

n – principal quantum numbern – determines permitted values of the energy

l = 0,1,2,…n-1;ml - magnetic quantum numberml – determines permitted values of the z-component of the orbital angular momentum

lml = ...,2,1,0

n = 1,2,3,4...

Wave functionsn,l,m

l = 1 m = ±1

l = 0n = 1

n = 2

n = 3

l = 0,1

l = 0,1,2

rerR ~)(

)()(,, , ll mmlnl rRr =

polynomial

~ ie

Quantum number notation

Degeneracy : one energy level En has different quantum numbers l and ml

l = 0 : s states n=1 K shell

l = 1 : p states n=2 L shell

l = 2 : d states n=3 M shell

l = 3 : f states n=4 N shell

l = 4 : g states n=5 O shell

. .

. .

1&10&1

1&10&0

2

======

===

l

l

l

l

mlmlmlml

n

0&01

===

lmln

2&21&20&2

1&22&2

1&10&1

1&10&0

3

==========

======

===

l

l

l

l

l

l

l

l

l

mlmlmlmlmlmlmlmlml

n

Electron states

1s

2s

2p

3s

3p

3d

M

L

K

S-states probability

P-states probability

Spin angular momentum and magnetic moment

Electron posseses spin angular momentum Ls. With this momentum magnetic momentum is connected:

se

es Lmeg

2

=

where ge is the gyromagnetic ratio

For free electron ge=2

se

s Lme

=

Allowed values of the spin angular momentum are quantized :

)1( = ssLs

spin quantum number s = ½ 23

=sL

Własny moment pędu - spin

The z – component of the spin angular momentum:

ssz mL =

=

2121

sm

Spin angular momentum and magnetic moment

To label completely the state of the electron in a hydrogen atom, 4 quantum numbers are need:

name label magnitude

Principal quantum number

n 1, 2, 3, ...

Orbital quantum number

l 0, 1, 2, ... n-1

magnetic quantum number

ml od –l do +l

Spin quantum number

ms ± 1/2

Many – electron atoms and the exclusion principle

Central field approximation: - Electron is moving in the total electric field due to the

nucleus and averaged – out cloud of all the other electrons. - There is a corresponding spherically symmetric potential –

energy function U( r). Solving the Schrodinger equation the same 4 quantum

numbers are obtained. However wave functions are different. Energy levels depend on both n and l.

• In the ground state of a complex atom the electrons cannot all be in the lowest energy state.

Pauli’s exclusion principle states that no two electrons can occupy the same quantum – mechanical state. That is, no two electrons in an atom can have the same values of all four quantum numbers (n, l, ml and ms )

Shells and orbitals

Nmax - maximum number of electrons occupying given orbital

n shell l orbital1 K 0 s2 L 0 s

L 1 p3 M 0 s

M 1 pM 2 d

4 N NNN

0123

spdf

Nmax

22

26

6

6210

1014

Shells K, L, M

n 1 2 3 0 0 1 0 1 2

m 0 0 -1 0 1 0 -1 0 1 -2 -1 0 1 2ms

N 2 8 18

N : number of allowed states state with ms = +1/2 state with ms = -1/2

­¯

­¯

­¯ ­ ­

­¯ ­ ­ ­¯

1s22s22p2

1s22s22p4

carbon

oxygen

Hund’s rule - electrons occupying given shell initially set up their spins paralelly

The periodic table of elements

Atoms of helium, lithium and sodium

n =1, = 0 n =1, = 0 n =1, = 0

n =2, = 0 n =2, = 0n =2, = 0

n =2, = 1 n =2, = 1

n =3, = 0

Helium (Z = 2) Lithium(Z = 3) Sodium (Z= 11)

1s

2s

2p

3s

1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6 6s2 4f14 5d10 6p6 7s2 6d10 5f14

110

25

15

23

22

21

26

162

43:

43:

43:

43:

43:

43:

43:

431:

sdCu

sdMn

sdCr

sdV

sdTi

sdSc

spCa

spsK

Electron configuration – the occupying of orbitals

Example: l = 1, s = ½

1= jjJ

21

21

23

21

21

23

21

21

23

21

, lub,,,

1lub1

==

====

jj mm

jj

j = 3/2 j = 1/2

SLLJ

=

Possible two magnitudes of j : l-sjslj == or

jjjjmmJ jjz ,1,,1,, ==

Total angular momentum - J

NMR ( nuclear magnetic resonance)Like electrons, protons also posses magnetic moment due to orbital angular momentum and spin ( they are also spin-1/2 particles) angular momentum.

Spin flip experiment:

Protons, the nuclei of hydrogen atoms in the tissue under study, normally have random spin orientations. In the presence of a strong magnetic field, they become aligned with a component paralell to the field. A brief radio signal flips the spins; as their components reorient paralell to the field, they emit signals that are picked up by sensitive detectors. The differing magnetic environment in various regions permits reconstruction of an image showing the types of tissue present.