i hbar d/dt(psi) = H psi - IC: t=0, psi(x, 0) = f(x) - BC: t->inf, psi->0 for + and -

Exam Prep Flashcards by Ai DontKnow

Planck-DeBroglie dispersion relation for particles

omega = hbar*k^2/(2m)

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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

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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

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Delta function: sifting property

Integral(-inf, inf) f(x)delta(x-x0) dx = integral(x0-, x0+) f(x0)delta(x-x0) dx = f(x0)

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Delta function: IBP

integral(-inf, inf) f(x) delta’(x-x0) dx = -f’(x0)

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Delta function: symmetry

Even function

Integral(0, inf) f(x)*delta(x) dx = 1/2 * f(0)

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Delta function: scaling

Delta(kx) = 1/|k| * delta(x)

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Delta function: composition

g(x) differentiable with simple zeroes at xi

Delta(g(x)) = sum(i) delta(x-xi)/|g’(xi)|

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Momentum operator in position representation

derived by computing <p> in position representation

p = hbar/i * d/dx</p>

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Phase velocity of wave packet

v = omega(k)/k

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Group velocity of wave packet

V = d/dk (omega(k))

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Dispersion relation for light wave

Omega = ck

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TDSE

ihbard/dt(psi) = H*psi

IC: t=0, psi(x, 0) = f(x)
BC: t->inf, psi->0 for + and -

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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

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Inner product

= integral(-inf, inf) f*(x)g(x) dx

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Orthogonality

= delta_mn

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Completeness

Sum(n) psi*(x’) * psi(x) = delta(x’-x) (position representation)
Integral(-inf, inf) |x>

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Transmission probability

Flux of stuff out / flux of stuff in

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Reflection probability

Flux of stuff out backwards / flux of stuff in same

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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

= bra(psi) a ket(psi)

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) i*hbar*L_k

Ladder operators

L+- = Lx +- iLy L+- Ylm = Clm+- Yl,m+-1

Hbar/2 * (1, 0 / 0, -1)

Hbar/2 * (0, 1 / 1, 0)

Hbar/(2i) * (0, 1 / -1, 0)

Degenerate perturbation theory

Hab = psia* x H1 x psib - compute the matrix - EVP: H1*c = E1*c 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