Quantum Mechanics Flashcards
integral limits of a function contained within|x|< b
-b to b
What happens if after measuring E1, you repeat the measurement on the same system
Measurement collapses the wavefunction and so repeated measurments will always yield E1
what is coulomb potential (V(r)) proportional to
1/r (and therefore equal to A/r)
what property of the energy levels of the harmonic oscillator follows from the commutator [H,a+] = hwa+
(assume h=h-bar)
the energy levels are equally spaced with spacing hw
radial probability distribution,
how do you find the the most probable value of r for an electron to be found
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
how to calculate first order correction (aka perturbation to energy) for all energy levels
given perturbation potential V’(x) and wavefuntion ψn(x).
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.
non-degenerate perturbation theory
En1/En0 = (1/En0)<ψn0|H’|ψn0>
potential energy of hydrogen
V(r) = -e^2/4π(epsilon)r
superposition of eigenstates, form of function
(time-dependent)
Ψ(x,t) = ψ1(x)e^(-iE1t/h) + ψ2(x)e^(-iE2t/h)
where h=h_bar
what is another way of writing |E|^2
E*E
how does the hamiltonian relate to energy
Hψn = Enψn
ψ1 * ψ2 dx = ?
(aka conjugate of 1 times regular 2, if they were the same n would equal 1)
0
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?
- 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
spherical coordinate system
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.
expectation value of energy
<ψ|H|ψ>
insert the hamiltonian into centre of integral.
H = (-h^2/2m) d^2/dx^2
(where h=h_bar)
schodinger equation
(-h^2/(2mr^2)) d/dr(r^2(dψ/dr)) + V(r)ψ = Eψ
(note: can ignore constants when subbing in ψ)
explain the significance of the quantum numbers n,l,m
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)
quality of states and angular momentum
states have non-zero angular momentum
what does the commutator [x_hat, xPz] equal
0, because P is not x based (if it was would equal xih_bar)
probability of E1
|c1|^2
En relation to E1
En = n^2 E1
what does the stern-gelach apparatus do?
equation for number of beams
produces a force on a beam of particles that depends on the z-component of the particles’ magnetic moment
beams split into = 2s+1
degeneracy of states general equation
n^2 sequence
eg. n=1, degeneracy = 1
n=2, degenracy = 4
angular momentum operator
J_hat = L_hat + S_hat
J^2 = j(j+1)h_bar
= L^2 + S^2 + 2S*L
semi-infinite well potential sketch key points, effect of slightly reducing V0
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
how to integrate powers of trig
cos^n(x)sin^m(x) dx
if n is odd, sub u=sinx
if m is odd, sub u=cosx
what is the result when an angular momentum operator is applied to its corresponding eigenfunction
scalar multiple of the same eigenfunction
how to integrate e^(iθ)
= -i e^(iθ)
from 0 to 2(pi), use euler:
e^(ix) = cos(x) +isin(x)
what to do when x_hat or p_hat in in question
times by x_bar = x or p_bar = -ih_bar(d/dx)
not <x> or <p>
no need to integrate</x>
energy of harmonic oscillator
En = hw((1/2) + n)
odd values of n give solution to potential
(where h = h_bar)
what does the imaginary part of the wavevector represent
decay of amplitude of wave
(for calculations of real things jusst take real part)
