Quantum Mechanics & Atomic Physics

Question 1

Expectation value of an observable

Answer 1

Question 2

Probability of finding a particle somewhere

Answer 2

Question 3

Hermitian operator facts

Answer 3

Question 4

position and momentum operators

Answer 4

Question 5

Expansion coefficient

Answer 5

Question 6

Expectation value in a superposition of states

Answer 6

Question 7

Definition of inner product behavior under complex conjugation of a bracket

Answer 7

Question 8

Applying an operator to a bracket

Answer 8

Question 9

Inner product on a function space

Answer 9

Question 10

Time dependent Schrödinger equation

Answer 10

Question 11

Hamiltonian operator

Answer 11

Question 12

Answer 12

Question 13

Commutator of the position and momentum operators

Answer 13

Question 14

Generalized uncertainty principle

Answer 14

Question 15

Variance

Answer 15

Question 16

Uncertainty of position and momentum

Answer 16

Question 17

“Folklore” uncertainty of x and p

Answer 17

ΔxΔp ≈ ħ

Question 18

“Folklore” uncertainty of energy and time

Answer 18

ΔEΔt ≈ ħ

Question 19

Hamiltonian for the quantum harmonic oscillator in one dimension

Answer 19

Question 20

Hamiltonian for the quantum harmonic oscillator in one dimension in terms of raising and lowering operators

Answer 20

Question 21

Commutation relation of the raising and lowering operators

Answer 21

Question 22

Spectrum of the harmonic oscillator

Answer 22

Question 23

Virial theorem

Answer 23

Question 24

Energy of a harmonic oscillator in 3 dimensions

Answer 24

Question 25

Wavefunction and energy of a free particle

Answer 25

Question 26

de Broglie relation

Answer 26

p = ℏk = h/λ

Question 27

Normalization of the radial part of the wavefunction

Answer 27

Question 28

Normalization of the angular part of the wavefunction

Answer 28

Question 29

Angular momentum operator in x

Answer 29

Question 30

Angular momentum operator in y

Answer 30

Question 31

Angular momentum operator in z

Answer 31

Question 32

Answer 32

Question 33

Answer 33

Question 34

Answer 34

Question 35

Answer 35

Question 36

Hamiltonian of the hydrogen atom

Answer 36

Question 37

Bohr radius

Answer 37

Question 38

Scale of the ground state hydrogen wavefunction

Answer 38

Question 39

Ground-state energy of a hydrogenic atom

Answer 39

Question 40

Excited-state energy of hydrogenic atom

Answer 40

Question 41

Fine structure constant

Answer 41

Question 42

de Broglie wavelength

Answer 42

λ = h / p

Question 43

Definition of a bra

Answer 43

Question 44

Bohr theory energy to remove an electron from a single-electron, multi-proton atom

Answer 44

E=Z²E₁

Question 45

Products of operators [AB,C]

Answer 45

A[B,C] + [A,C]B

Question 46

Products of operators [A,BC]

Answer 46

[A,B]C + B[A,C]

Question 47

Raising/lowering normalization factors

Answer 47

Question 48

Δt

Answer 48

The amount of time it takes the expectation value of Q to change by one standard deviation

If any observable changes rapidly, the uncertainty in energy must be large.

Question 49

Answer 49

Question 50

Answer 50

Question 51

Spin raising and lower operators

Answer 51

Question 52

Blackbody power spectrum

Answer 52

Question 53

Stefan-Boltzmann law

Answer 53

dP / dA ∝ T⁴

Question 54

Wavelength of the peak of a blackbody spectrum, given by Wien’s displacement law

Answer 54

Question 55

L_z

Answer 55

Question 56

S_z

Answer 56

Question 57

Answer 57

Question 58

Answer 58

Question 59

Answer 59

Question 60

Range of possible values for L_z

Answer 60

Question 61

Number of allowed values for L_z

Answer 61

2l+1

Question 62

Range of possible values for S_z

Answer 62

Question 63

Range of possible values for J_z

Answer 63

Question 64

Range of possible values for L_q

Answer 64

Question 65

Range of possible values for S_q

Answer 65

Question 66

Range of possible values for J_q

Answer 66

Question 67

Selection rules for Δn

Answer 67

Δn = anything

Question 68

Selection rules for Δl

Answer 68

Δl = ±1

Question 69

Selection rules for Δm_l

Answer 69

Δm_l = 0, ±1

Question 70

Selection rules for Δm_j

Answer 70

Δm_j = 0, ±1

Question 71

ΔS’ selection rules for multi-electron atoms

Answer 71

ΔS’ = 0

Question 72

Special selection rule for J of multi-electron atoms

Answer 72

Question 73

Energies for particle in a box

Answer 73

Question 74

Number of electrons in each full subshell

Answer 74

(2s+1)(2l+1)

2(2l+1) when s=1/2

Question 75

Energy of a particle in an infinite box

Answer 75

Question 76

Number of extrema in a harmonic oscillator wavefunction

Answer 76

n+1

Question 77

Symmetry of a wavefunction with odd n

Answer 77

anti-symmetric

Question 78

Symmetry of a wavefunction with even n

Answer 78

symmetric