Download m L

Today’s Lecture
●Spatial Quantisation
●Angular part of the wave function
●Spin of the electron
●Stern – Gerlach experiment
●Internal magnetic fields in atoms
●Resulting Fine Structure.
Spatial Quantization and Electron Spin
The angular wavefunctions for the H
atom are determined by the values of l
and ml. Analysis of the wavefunctions
shows that they all have
Angular Momentum Z
given by l
and projections
L
onto the z-axis
ml 
of L of
| L | l (l  1)
Lz= mlh/2p.
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 out
of 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)
The picture of a precessing vector for L
helps to visualise the results
(2l + 1 )orientations
in general
Krane p216
This is another manifestation
Of the Uncertainty Principle
LZ .  h/2p
Product of Radial and Angular Wavefunctions
n=1 spherical
n=2, l=0
spherical, extra
radial bump
n=2, l=1, ml= 1
equatorial
n=2, l=1,
ml=0 polar
n=3
spherical
for l=0
l=1,2
equatorial
or polar
depending
on ml.
Krane p219
 2 for different sets of quantum numbers
The Z axis is in the vertical direction.
L and the dipole moment
Classical-electron in
orbit
Z
Quantum system is
Spatially quantised
Electric dipole
Magnetic dipole
STERN-GERLACH Apparatus
1h
L =  l(l + 1) h
0
-1h
We would expect that the beam splits
in three on passing through the
Inhomogeneous field.
If l = 0 we expect only one image.
Beam of Ag ions used. L = 0,1,2,3,-----Hence odd no. of images expected.
In practice L = 0 but it does not really matter-the
Main point is that we should have an odd no. of
Images.
(1921)
Electron Spin
Electrons have an intrinsic spin which we will see is
also spatially quantized (just as we have seen the
orbital angular momentum to be 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 s =1/2 and ms= 1/2 for electrons.
Magnetic Fields Inside Atoms
Moving charges are electrical currents and hence create
magnetic fields.
Thus, there are internal magnetic 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.
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
ms anti-aligned (ms = spin magnetic dipole moment).
For electrons we have :
(the minus sign arises since
the electron charge is negative)
µS = – (e/m) S
The Zeeman effect
As we saw from the Stern-Gerlach experiment:
Electron levels (with l different from zero) split in
external magnetic field
1h
For simplicity we ‘forget’ about spin for now
L =  l(l + 1) h
0
-1h
We would expect that the beam splits
in three on passing through the
Inhomogeneous field.
If l = 0 we expect only one image.
Moving charge => magnetic moment
L
r
m
ei
e
e
e
2
m L  iA 
vpr 
vmr  
L
2pr
2m
2m
The -ve sign indicates that the
vectors L and mL point in
opposite directions.
The z-component of the magnetic moment:
m L, z
e
e

Lz  
ml    ml m B
2me
2me
e
where m B 
 9.274 x 10  24 J / T
2me
The z-component of mL is given in units
of the Bohr magneton, mB
Estimation of the Zeeman splitting
U   m L  B   m L , z B  ml m B B
l=1, ml=+1
U
l=1
l=1, ml=0
U
l=1, ml= -1
For B=1 T (quite large external magnetic field):
Splitting U=6x10-6 eV (small compared to
the eV energies of the lines
Selection rule: Δml=0,±1
Magnetic Fields Inside Atoms
Moving charges are electrical currents and hence create
magnetic fields.
Thus, there are internal magnetic 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.
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 ms
anti-aligned (ms = spin magnetic dipole moment).
For electrons we have :
(the minus sign arises since the
electron charge is negative)
µS = – (e/m) S
Fine structure
Even without an external magnetic field there
is a splitting of the energy levels (for l not zero).
It is called fine structure.
The apparent movement of the proton
creates an internal magnetic field
Magnetic Fields Inside Atoms
Moving charges are electrical currents and hence create
magnetic fields.
Thus, there are internal magnetic 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 magnetism, µ=iA for a current loop of area A, so
in the Bohr model the magnetic moment, m , is
q
q
q 
2
m  iA 
pr 
rp 
|L|
2prme p 
2me
2me

e 
Thus, if q  e, m L  
L
2me
Electron Spin
Electrons have an intrinsic spin which we will see is also
spatially quantized
(just as we have seen the orbital angular momentum to be
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 s =1/2 and ms= 1/2 for electrons.
L
The -ve sign indicates that the
vectors L and mL point in
opposite directions.
r
m
ei
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.