Quantum Mechanics

Question 1

integral limits of a function contained within|x|< b

Answer 1

-b to b

Question 2

What happens if after measuring E1, you repeat the measurement on the same system

Answer 2

Measurement collapses the wavefunction and so repeated measurments will always yield E1

Question 3

what is coulomb potential (V(r)) proportional to

Answer 3

1/r (and therefore equal to A/r)

Question 4

what property of the energy levels of the harmonic oscillator follows from the commutator [H,a+] = hwa+
(assume h=h-bar)

Answer 4

the energy levels are equally spaced with spacing hw

Question 5

radial probability distribution,
how do you find the the most probable value of r for an electron to be found

Answer 5

P(r) dr = r^2R(r)^2 dr
(techincally proportional to not equals, so can ignore constants given with R(r))

electron is most likely to be found at dP/dr = 0

Question 6

how to calculate first order correction (aka perturbation to energy) for all energy levels

given perturbation potential V’(x) and wavefuntion ψn(x).

Answer 6

En = <ψn|V’(x)|ψ>
(intregral between given limits for ψ squared times V)

Results in a funtion concerning n. Determine solution for even and odd n to find correction to all levels.

Question 7

non-degenerate perturbation theory

Answer 7

En1/En0 = (1/En0)<ψn0|H’|ψn0>

Question 8

potential energy of hydrogen

Answer 8

V(r) = -e^2/4π(epsilon)r

Question 9

superposition of eigenstates, form of function
(time-dependent)

Answer 9

Ψ(x,t) = ψ1(x)e^(-iE1t/h) + ψ2(x)e^(-iE2t/h)
where h=h_bar

Question 10

what is another way of writing |E|^2

Answer 10

E*E

Question 11

how does the hamiltonian relate to energy

Answer 11

Hψn = Enψn

Question 12

ψ1 * ψ2 dx = ?
(aka conjugate of 1 times regular 2, if they were the same n would equal 1)

Answer 12

0

Question 13

for a finite potential well with a lower potential on the left than right, what do the wavefunction sketch’s look like for E1 and E2?

Answer 13

• E1 has 1 turning point, E2 has 2
• functions should decay to zero more gradually on the left than the right (there should be bleeding out the fucntion on both sides)
• turning points should be slightly displaced to the left relative to the symmetric case

Question 14

spherical coordinate system

Answer 14

r^2 sinθ dθdφdr

x = rcosθcosφ
y = rsinθsinφ
z = rcosθ

typically, integrate θ between pi and 0, φ between 2pi and 0, and r between infinity and 0.

Question 15

expectation value of energy

Answer 15

<ψ|H|ψ>
insert the hamiltonian into centre of integral.
H = (-h^2/2m) d^2/dx^2

(where h=h_bar)

Question 16

schodinger equation

Answer 16

(-h^2/(2mr^2)) d/dr(r^2(dψ/dr)) + V(r)ψ = Eψ

(note: can ignore constants when subbing in ψ)

Question 17

explain the significance of the quantum numbers n,l,m

Answer 17

n: principal/radial, determines energy of the state
l: total angular momentum
m: magnetic, labels the z-component of the angular momentum

l<n, -l</=m</=l (</= less than or equal to)

Question 18

quality of states and angular momentum

Answer 18

states have non-zero angular momentum

Question 19

what does the commutator [x_hat, xPz] equal

Answer 19

0, because P is not x based (if it was would equal xih_bar)

Question 20

probability of E1

Answer 20

|c1|^2

Question 21

En relation to E1

Answer 21

En = n^2 E1

Question 22

what does the stern-gelach apparatus do?
equation for number of beams

Answer 22

produces a force on a beam of particles that depends on the z-component of the particles’ magnetic moment

beams split into = 2s+1

Question 23

degeneracy of states general equation

Answer 23

n^2 sequence
eg. n=1, degeneracy = 1
n=2, degenracy = 4

Question 24

angular momentum operator

Answer 24

J_hat = L_hat + S_hat
J^2 = j(j+1)h_bar
= L^2 + S^2 + 2S*L

Question 25

semi-infinite well potential sketch key points, effect of slightly reducing V0

Answer 25

0 @ x=0, continuous positve gradient @ x=L,
peaks/troughs shifted slightly to right compared to infinite well,
exponential decay of excited states is longer/slower than ground state.

if V0 is reduced, wavelength increases with much larger decay. Peaks/troughs stay within L

Question 26

how to integrate powers of trig
cos^n(x)sin^m(x) dx

Answer 26

if n is odd, sub u=sinx
if m is odd, sub u=cosx

Question 27

what is the result when an angular momentum operator is applied to its corresponding eigenfunction

Answer 27

scalar multiple of the same eigenfunction

Question 28

how to integrate e^(iθ)

Answer 28

= -i e^(iθ)
from 0 to 2(pi), use euler:
e^(ix) = cos(x) +isin(x)

Question 29

what to do when x_hat or p_hat in in question

Answer 29

times by x_bar = x or p_bar = -ih_bar(d/dx)
not <x> or <p>
no need to integrate</x>

Question 30

energy of harmonic oscillator

Answer 30

En = hw((1/2) + n)
odd values of n give solution to potential

(where h = h_bar)

Question 31

what does the imaginary part of the wavevector represent

Answer 31

decay of amplitude of wave

(for calculations of real things jusst take real part)