Download 1AMQ, Part II Quantum Mechanics

1AMQ, Part III The Hydrogen Atom
5 Lectures
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Spectral Series for the Hydrogen Atom.
Bohr’s theory of Hydrogen.
The Hydrogen atom in quantum
mechanics.
Spatial quantization and electron spin.
Fine Structure and Zeeman splitting.
•K.Krane, Modern Physics, Chapters 6 and 7
• Eisberg and Resnick, Quantum Physics,
Chapters 4, 7 & 8
1AMQ P.H. Regan &
W.N.Catford
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Atomic Line Spectra
Light emitted by free atoms has fixed or discrete
wavelengths. Only certain energies of photons
can occur (unlike the continuous spectrum
observed from a Black Body).
Atoms can absorb energy (become excited) by
collisions, fluorescence (absorption and reemission of light) etc. The emitted light can be
analysed with a prism or diffraction grating with
a narrow collimating slit (see figure below).
The dispersed
image shows a
series of lines
corresponding to
different
wavelengths,
called a line
spectrum. (note
you can have both
emission and
absorption line
spectra)
1AMQ P.H. Regan &
W.N.Catford
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Hydrogen-The Simplest Atom
Atomic hydrogen (H) can be studied in gas discharge
tubes. The strong lines are found in the emission
spectrum at visible wavelengths, called Ha, Hb and
Hg. More lines are found in the UV region, more
which get closer and closer until a limit is reached.
1AMQ P.H. Regan &
W.N.Catford
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The visible spectrum lines are the Balmer Series.
Balmer discovered that the wavelengths of these
lines could be calculated using the expression,
c
1
1
n 
 cRH ( 2  2 ), n  3,4,5....
n
2 n
where, RH= the Rydberg constant for hydrogen
= 1.097x107m-1 = 1/911.76 angstroms
Balmer proposed more series in H with wavelengths
given by the more general expression,
c
1
1
 n   cRH ( 2  2 )
n
n2 n1
n1>n2 for
positive
integers.
These series are observed experimentally and
have different names n2= 1 Lyman Series (UV)
= 2 Balmer Series (VIS)
= 3 Paschen Series (IR)
= 4 Bracket Series (IR)
= 5 Pfund Series (IR)
1AMQ P.H. Regan &
W.N.Catford
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1AMQ P.H. Regan &
W.N.Catford
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These lines are also seen in stellar spectra from
absorption in the outer layers of the stellar gas.
1AMQ P.H. Regan &
W.N.Catford
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The Rydberg-Ritz Combination Principle is an
empirical relationship which states that if
n1 and n2 are any 2 lines in one series, then
|n1n2| is a line in another series.
Electron Levels in Atoms.
Discrete wavelengths for emis/abs. lines
suggests discrete energy levels. Balmer’s
formula suggests that the allowed energies
are given by (cRH / n2) (for hydrogen).
A more detailed study is possible
using controlled energy collisions
between electrons and atoms such as
the Franck-Hertz experiment.
1st excited state
e-s with enough
Excitation
energy can cause
Energy
this transition.
1AMQ P.H. Regan &
W.N.Catford
ground state,
(ie. lowest one)
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1AMQ P.H. Regan &
W.N.Catford
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Bohr Theory: Bohr’s postulates defined a
simple ordered system for the atom.
1AMQ P.H. Regan &
W.N.Catford
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h
since, vn me rn  Ln  n
 n
2
me n 2
Ze 2
then,
.(
) 
rn me rn
40 rn2
2
4


2
0
and hence, rn  n .a0 where, a0 
2
me Ze
a0 is the Bohr radius (ie. smallest allowed)
For hydrogen, Z=1 and a0=0.529Angstroms, ie
model predicts ~10-10m for atomic diameter.
For these allowed radii, we can calculate the
allowed energies of the levels in the Bohr atom
2
1
Ze
2
En  K  V  me vn 
2
40 rn
me vn2
Ze 2
and

for mech. stab.
2
40 rn
rn
1 Ze 2
thus, En   .
 K
2 40 rn
1AMQ P.H. Regan &
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Thus by substitution, we obtain
 n 

Ze  4 0m v r  4 0me rn .
 me rn 
2 2
n

2
ie.
Ze  40
me rn
2
2
2
e n n
2
2
n
thus, rn  40
, n  1,2,3,4...
2
me Ze
n
1 Ze 2
and vn 

, n  1,2,3...
me rn 40 n
by substituti on for rn , we obtain,
me Z 2 e 4 1
En  
. 2 , n  1,2,3.....
2
(40 )2 n
ie. quantization of the angular momentum
leads in the Bohr model to a quantization of
the allowed energy states of the atom.
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For H, the ground state energy is given by
me Z 2 e 4 1
n  1, thus, E1  
. 2
2
(40 )2 1
ie.
E1  2.18x10 -18 J  13.6 eV
13.6
and En   2 eV
n
1AMQ P.H. Regan &
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n
2.19x106 ms 1
vn 

2
me n a0
1
ie. for n  1, H, b  0.73%  1 (non - rel).
The allowed energies let us calculate the
allowed frequencies for photons emitted in
transitions between different atomic levels
ie.
hn  Einitial-Efinal
If ni and nf are the quantum numbers of the
initial and final states, then, for H, if nf < ni
 1

13.6eV  1
1 
1
n
. 2  2  cRH . 2  2 
n n 
n n 
h
f 
f 
 i
 i
me e 4
1
thus, RH 

2 3
4 (40 )  c 911.27 Angs
This prediction of the Bohr model
compares with an expt. value of
1/911.76Angs, ie. accurate to with 0.05%
in H. (Exact agreement if motion of
nucleus is included. ie nucleus and electron
move around the atoms centre of mass.)
Balmer series correspond
to when nf=2
1AMQ P.H. Regan &
W.N.Catford
Lyman series correspond
to when nf=1
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The Lyman series correspond to the highest energy
(shortest wavelength) transitions which H can emit.
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Deficiencies of the Bohr Model
• No proper account of quantum
mechanics (de Boglie waves etc.)
• It is planar and the `real world’ is
three dimensional .
• It is for single electron atoms only.
• It gets all the angular momenta
wrong by one unit of h/2.
1AMQ P.H. Regan &
W.N.Catford
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The Hydrogen Atom in Quantum Mechanics
The e- is bound to the nucleus (p) by the Coulomb
pot. This constraint leads to energy quantization. The
Time Independent Schrodinger Equation can be used.
For hydrogen, (Z=1)
2  d 2
d2
d2 
 2  2  2   V ( x, y, z )  E

2m  dx
dy
dz 
1
e2
where, V ( x, y, z )  
.
40 x 2  y 2  z 2
It is easier to solve this in spherical polar rather
than cartesian coordinates, this we have
2  d 2 2 d
1
d
1
d2 
 2 


 2
(sin 
) 2 2
2 
2m  dr
r dr r sin 
d
r sin  d 
1 e2
 V (r , ,  )  E , where V (r , ,  )  
40 r

This equation is said to be separable ie,
 (r , ,  )  R(r ).( ).( )
1AMQ P.H. Regan &
W.N.Catford
z r y
x
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R(r )  Radial Wavefunti on
( ). ( )  Angular Wa vefunction s
The three spatial dimensions (r,,) lead to 3
quantum numbers, which relate to
• How far the orbital is from the nucleus (n)
• How fast the orbit is (ie. angular momentum) (l)
• Then angle of the orbit in space (ml).
The quantum numbers and their allowed
values are
• n principle quantum number, 1,2,3,4,5…
• l angular momentum q.n. 0,1,2,3,4..(n-1)
• ml magnetic q.n. -l-l+1,...-1,0,1,….l-1,l
1AMQ P.H. Regan &
W.N.Catford
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1AMQ P.H. Regan &
W.N.Catford
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Energies and Degeneracies:
Each solution, n,l,ml has an energy that
depends only on n (En=Bohr value) and
there are n2 solutions (ie. all the possible
values of l and ml ) for each energy En.
Radial Wavefunctions.
Determined by n (main factor in
determining the radius) and l measures
the electrons angular momentum.
If L is the angular momentum vector,
then
| L | l (l  1).
| Rn ,l (r ) |2  prob. for small volume at r , , 
for all  ,  ,
r 2 | Rn ,l (r ) |2
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W.N.Catford
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W.N.Catford
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W.N.Catford
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Spatial Quantization and Electron Spin
The angular wavefunctions for the H
atom are determined by the values of l
and ml. Analysis of the w.functions
shows that they all have Ang. Mom.
given by l. and
Z
projections onto
L
the z-axis of L of
Lz= mlh/2.
m
l
| L | l (l  1)
Quantum mechanics says that only certain
orientations of the angular momentum are
allowed, this is known as spatial quantization.
For l=1, ml=0 implies an axis of rotation in
the x-y plane. (ie. e- is out of x-y plane),
ml = +1 or -1 implies rotation around Z (e- is
in or near x-y plane)
1AMQ P.H. Regan &
W.N.Catford
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The picture of a precessing vector for L
helps to visualise the results
Krane p216
1AMQ P.H. Regan &
W.N.Catford
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Product of Radial and Angular Wavefunctions
l=1, ml=+-1 equatorial
n=1 spherical
l=1, ml=0 polar
n=1, l=0 spherical,
extra radial bump
n=3
spherical
for l=0
l=1,2
equatorial
or polar
depending
on ml.
Krane p219
1AMQ P.H. Regan &
W.N.Catford
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Magnetic Fields Inside Atoms
Moving charges are currents and hence create
magn. fields. Thus, there are internal B- fields in
atoms. Electrons in atoms can have two spin
orientations in such a field, namely ms=+-1/2….and
hence two different energies. (note this energy
splitting is small ~10-5 eV in H). We can estimate the
splitting using the Bohr model to estimate the internal
magnetic field, since
In the Bohr model the magnetic moment, m , is 
q
q
q
2
m  iA 
r 
rp 
|L|
2rme p )
2me
2me

e 
Thus, if q  e, m L  
L
2me
Electron Spin
Electrons have an intrinsic spin which is also
spatially quantized.
Spinning charges behave like dipole magnets.
The Stern-Gerlach experiment uses a
magnetic field to show that only two projections
of the electron spin are allowed. By analogy
with the l and ml quantum
numbers, we see that
1AMQ P.H. Regan &
W.N.Catford
S=1/2 and ms=+-1/2 for electrons.
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The -ve sign indicates that the
vectors L and mL point in
opposite directions.
L
r
e-
m
i
The z-component of mL is given in units
of the Bohr magneton, mB, where
m L, z
e
e

Lz  
ml   ml m B
2me
2me
e
where m B 
 9.274 x 10  24 J / T
2me
 

E   m s .Bint where m s  spin magnetic moment

e 
and m s   S (spinning charge acts like magnet)
m
This energy shift is determined by the
relative directions of the L and S vectors.
1AMQ P.H. Regan &
W.N.Catford
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Fine Structure and Zeeman Splitting
For atomic
electrons, the
relative orbital
motion of the
nucleus creates
a magnetic field
(for l=0).
The electron
spin can have
ms=+-1/2
relative to the
direction of the
internal field,
Bint.
The state with
ms aligned with
Bint has a lower
energy than
when antialigned.
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W.N.Catford
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Spin-Orbit Coupling
For a given e-, L and S add together such that J=L+S

| J |
j ( j  1) , j  l  12
1AMQ P.H. Regan &
W.N.Catford
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Atomic Doublets
Levels with l=0 are split into energy doublets,
called fine structure, due to spin-orbit coupling.
The fine structure is approx. 10-5eV in Hydrogen
and increases as Z4 for heavier elements.
We can use the Bohr model get an estimate of
the spin-orbit splitting, by assuming an electron
orbit of radius r, carrying current i establishing
a magnetic field, B at the centre of loop. Thus,
m 0i m 0 e m 0 ev
m 0 ev
B


and E  2 m B B 
mB
2
2r 2rT 2r 2r
2r

n
In the Bohr model, | L | me vr  n and v 
,
me r
m0e 2 2 n
thus we have E 
4me2 r 3
substituti ng in for the Bohr radius (r  a0 ) and
recalling that c 2 
1
 0 m0
, then, E  (me c 2 )a 4
1
n2
e2
1
where a 
 fine structure constant 
40 c
137
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Zeeman Effect
If the atom is placed in an external magnetic
field, the e- orbital angular momentum (l) can
align with the field direction.While this
magnetic field is `switched on’ there will be an
extra splitting of the energy levels (for l=0).
e 
since, m L  
L, energy shift is
2me
 
E   m L .B, thus the new levels have
e  
E B0  E  E B0 
L.B
2me

Thus, is Z  direction of B - field, then
e 
E B0  E  E B0  ml
|B|
2me
ie. splitting is given by E  ml m B B
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Splitting is further affected by ms=+-1/2
n=1, l=1, j=1/2 splits into mj=-1/2, +1/2
n=1, l=1, j=3/2 splits into mj=-3/2,-1/2,+1/2,+3/2
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