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1

Orbital Angular Momentum Operator

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2

Laplacian in spherical polar coordinates

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3

Orbital Angular Momentum Operator in spherical coordinates

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4

Spherical polar coordinates

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5

Spherical polar coordinates

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6

Spherical polar coordinates

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7

Spherical polar coordinates

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8

Eigenvalues and eigenfunctions are given by

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9

z-component is given by

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10

Both L^2 and L^z assume

one set of discrete values

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11

In terms of p^ and r^

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12

In terms of p^ and r^

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13

In terms of p^ and r^

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17

L^2 commutes with

L^x, L^y, and L^z and L^+ and L^-

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18

Raising Operator

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Lowering Operator

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20

Raising and lowering operators are not

Hermitian

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21

Raising and lowering operators do not

represent physical quantities

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22

Raising and lowering operators do

raise the eigenfunctions of L^z up or down the â€śladderâ€ť of eigenvalues L^z

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25

What is the effective current, i, for a particle with charge e-, mass me, moving in an orbit at a radius of a0 at an angular velocity Ď‰?

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26

Magnetic moment associated with effective current, i, for a particle with charge e-, mass me, moving in an orbit at a radius of a0 at an angular velocity Ď‰

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27

Orbital angular momentum, **L**

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gyromagnetic ratio

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31

Bohr magneton

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32

What happens when you apply a **B** field?

The energy levels split by Î”E

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33

Î”E

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34

What did Stern Gerlach experiment find?

An atomâ€™s orbital angular momentum is quantised

some particles possess an intrinsic angular momentum, called spin

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35

What is spin, S?

Intrinsic angular momentum, possessed by some particles (e.g. electrons, protons, neutrons)

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36

What are the spin quantum numbers for an electron?

s = 1/2

ms = +/- 1/2

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37

Does orbital angular momentum require that there is an odd or even number of eigenstates of L^z?

Odd â†’ 2l+1

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42

**Orbital** angular momentum has an extra (?) condition that the wave function must be a well behaved function of position. What does this require?

n must be even and so l must have an integer value

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43

Does an electron spin?

No but it has intrinsic angular momentum which is called spin

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