Today’s Lecture ●Spatial Quantisation ●Angular part of the wave function ●Spin of the...

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