today’s lecture ●spatial quantisation ●angular part of the wave function ●spin of the...
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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 SpinThe angular wavefunctions for the H atom are determined by the values of l and ml. Analysis of the wavefunctionsshows that they all have Angular Momentumgiven by land projections onto the z-axis of L of Lz= mlh/2
L
lm)1(|| llL
Z
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
Krane p216
This is another manifestationOf the Uncertainty Principle
LZ . h/2
(2l + 1 )orientations in general
Krane p219
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=3
spherical for l=0
l=1,2 equatorial or polar depending on ml.
n=2, l=1, ml=0 polar
2 for different sets of quantum numbers
The Z axis is in the vertical direction.
L and the dipole moment
Electric dipoleMagnetic dipole
Classical-electron in orbit
Quantum system isSpatially quantised
Z
STERN-GERLACH Apparatus
1 h
0
-1h
L = l(l + 1) h
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-theMain point is that we should have an odd no. ofImages.
(1921)
Electron SpinElectrons 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.
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.
Magnetic Fields Inside Atoms
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 s aligned with Bint has a lower energy than s anti-aligned (s = spin magnetic dipole moment).
For electrons we have : (the minus sign arises since the electron charge is negative)
µS = – (e/m) S
1 h
0
-1h
L = l(l + 1) h
We would expect that the beam splits in three on passing through the Inhomogeneous field.If l = 0 we expect only one image.
As we saw from the Stern-Gerlach experiment:
Electron levels (with l different from zero) split in external magnetic field
The Zeeman effect
For simplicity we ‘forget’ about spin for now
The -ve sign indicates that the vectors L and L point in opposite directions.
The z-component of L is given in units of the Bohr magneton, B
TJm
e
mmm
eL
m
e
e
Blle
ze
zL
/10 x 274.92
where
22
24B
,
rL
e-
i
Moving charge => magnetic moment
Lm
evmr
m
erv
r
eiAL 222
2
The z-component of the magnetic moment:
Estimation of the Zeeman splitting
BmBBU BlzLL ,
l=1
l=1, ml=+1
l=1, ml=0
l=1, ml= -1
U
U
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
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 s aligned with Bint has a lower energy than s anti-aligned (s = spin magnetic dipole moment).
For electrons we have : (the minus sign arises since the
electron charge is negative)
µS = – (e/m) S
Magnetic Fields Inside Atoms
Fine structureEven 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
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, soin the Bohr model the magnetic moment, , is
Magnetic Fields Inside Atoms
Lm
eeq
Lm
qrp
m
qr
prm
qiA
eL
eee
2 , if Thus,
||222
2
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.
magnet) like acts charge (spinning and
moment magneticspin where. sint
Sm
e
BE
s
s
This energy shift is determined by the relative directions of the L and S vectors.
The -ve sign indicates that the vectors L and L point in opposite directions.
The z-component of L is given in units of the Bohr magneton, B, where
TJm
e
mmm
eL
m
e
e
Blle
ze
zL
/10 x 274.92
where
22
24B
,
rL
e-
i