Exam Prep Flashcards
Planck-DeBroglie dispersion relation for particles
omega = hbar*k^2/(2m)
Free particle FT pair
Psi(x, t) = 1/sqrt(2pi) * integral(-inf, inf) phi(k, t) e^(ikx) dk
Phi(k, t) = 1/sqrt(2pi) * integral(-inf, inf) psi(x, t) e^(-ikx) dx
Dirac delta function definitions (6)
1) integral(-inf, inf) f(x) delta(x-x0) dx = f(x0)
2) delta_n(x) = n for |x| <= 1/(2n)
3) delta_n(x) = n/sqrt(pi) * e^(-n^2x^2)
4) delta_n(x) = n/pi * 1/(1 + n^2x^2)
5) delta_n(x) = sin(nx)/(pi*x) = 1/(2pi) * integral(-n, n) e^(ikx) dk
6) delta(x) = 0 if x=/=0, undefined otherwise such that integral(-inf, inf) = integral(0-, 0+) = 1
Delta function: sifting property
Integral(-inf, inf) f(x)delta(x-x0) dx = integral(x0-, x0+) f(x0)delta(x-x0) dx = f(x0)
Delta function: IBP
integral(-inf, inf) f(x) delta’(x-x0) dx = -f’(x0)
Delta function: symmetry
Even function
Integral(0, inf) f(x)*delta(x) dx = 1/2 * f(0)
Delta function: scaling
Delta(kx) = 1/|k| * delta(x)
Delta function: composition
g(x) differentiable with simple zeroes at xi
Delta(g(x)) = sum(i) delta(x-xi)/|g’(xi)|
Momentum operator in position representation
- derived by computing <p> in position representation
p = hbar/i * d/dx</p>
Phase velocity of wave packet
v = omega(k)/k
Group velocity of wave packet
V = d/dk (omega(k))
Dispersion relation for light wave
Omega = ck
TDSE
ihbard/dt(psi) = H*psi
- IC: t=0, psi(x, 0) = f(x)
- BC: t->inf, psi->0 for + and -
TDSE solution method
1) separation of variables: x is TISE, theta(t) = exp(-i/hbarEnt)
2) linear superposition of solutions
3) BCs to get Fourier coefficients for constants Cn
Inner product
= integral(-inf, inf) f*(x)g(x) dx
Orthogonality
= delta_mn
Completeness
Sum(n) psi*(x’) * psi(x) = delta(x’-x) (position representation)
Integral(-inf, inf) |x>
Transmission probability
Flux of stuff out / flux of stuff in
Reflection probability
Flux of stuff out backwards / flux of stuff in same
Finding a finite discontinuity in psi’(x)
- when U(x) has an infinite jump
- integrate TISE from the inf jump left to inf jump right (ex, 0- to 0+)
Transmission through N identical replicas with period L
(AN, BN) = P^N (A0, B0) where P=DM
Eigenfunctions for SHO
Solutions to hermite’s DE
Def: Hilbert space
Linear vector space over the field of complex numbers C whose elements are complex-valued functions f(x) of real variable x, which are square integrable
Hilbert space properties (7)
1) linear
2) inner product defined as normal
3) norm of |f> = ||f|| as normal
4) completeness: “H contains all its limit points”
5) H is N-dimensional iff there exist N linearly indep vectors but no N+1 vectors are linearly independent
6) orthonormal basis can be generated via Gram-Schmidt orthogonalization procedure
7) in infinite dimensional H, we require completeness of the orthonormal basis set
Def: orthonormal basis is complete
There exists cn such that partial sums sum(n, N) cn |psi> converge to |f> in the mean lim(N->inf) ||f - sum(n, N) cn*psin|| = 0
Def: expectation value
<a> = bra(psi) a ket(psi)</a>
Def: hermitian operator
A* = A
=
Formal statement for probabilities
The only result of a measurement of the observable A, performed on a system in state |f>, is one of the eigvals An of the operator A
The probability of measuring the value of An is given by |psi* * f|^2 where |psin> is the eigvect corresponding to An obtained by solving eigval problem
Def: non-degenerate
Iff Am = An -> |psim> = |psin>
Def: N_n fold degenerate
N_n linearly independent eigvects |psi_nj> such that A|psi_nj> = An|psi_nj> for j=1, 2, …, N
Property of hermitian operator A
Eigvals are real
Eigfunctions are orthogonal (non-degenerate eigvals)
Superposition principle
Every physical state |f> can be represented by a generalized Fourier series
Properties of commutator
- independent of representation
- A and B share the same set of eigenvectors iff [A, B] = 0
——> sharing eigenvectors means uncertainties vanish in simultaneous measurements - when proving stuff don’t forget abt the non-degenerate vs degenerate cases
Angular momentum in 3D
Lx = yp(z) - zp(y) Ly = zp(x) - xp(z) Lz = xp(y) - yp(x)
Spherical coordinates
r = sqrt(x^2 + y^2 + z^2) Theta = arccos(z/r) Phi = arctan(y/x)
Spherical harmonics orthonormality relation
= delta_m’m*delta_l’l
Effect of magnetic field
Add a term to Hamiltonian: -mu*B, mu is magnetic moment
Angular momentum: commutator relationships
[Li, Lj] = epsilon(ijk) ihbarL_k
Ladder operators
L+- = Lx +- iLy
L+- Ylm = Clm+- Yl,m+-1
Sz
Hbar/2 * (1, 0 / 0, -1)
Sx
Hbar/2 * (0, 1 / 1, 0)
Sy
Hbar/(2i) * (0, 1 / -1, 0)
Degenerate perturbation theory
Hab = psia* x H1 x psib
- compute the matrix
- EVP: H1c = E1c where E1 = perturbation
Parity operator properties
- [A, P] = 0 -> eigenfunctions of A are either even or odd, and if psi1 and psi2 are diff parities, = 0 (A is even)
- {A, P} = AP + PA = 0 -> if psi1 and psi2 are same parity, then =0 (A is odd)
What does it mean for a free particle wave to propagate without distortion?
Psi(x, t) = f(x-vt) where v is constant
Debroglie wavelength relationship thing
k = p/hbar
Only one bound state
Only one k value for E < U
Quantized energy
Multiple kn
