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AMATH 373 > Exam Prep > Flashcards

Exam Prep Flashcards

1
Q

Planck-DeBroglie dispersion relation for particles

A

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

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2
Q

Free particle FT pair

A

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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3
Q

Dirac delta function definitions (6)

A

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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4
Q

Delta function: sifting property

A

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

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5
Q

Delta function: IBP

A

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

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6
Q

Delta function: symmetry

A

Even function

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

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7
Q

Delta function: scaling

A

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

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8
Q

Delta function: composition

A

g(x) differentiable with simple zeroes at xi

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

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9
Q

Momentum operator in position representation

A
  • derived by computing <p> in position representation

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

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10
Q

Phase velocity of wave packet

A

v = omega(k)/k

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11
Q

Group velocity of wave packet

A

V = d/dk (omega(k))

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12
Q

Dispersion relation for light wave

A

Omega = ck

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13
Q

TDSE

A

ihbard/dt(psi) = H*psi

  • IC: t=0, psi(x, 0) = f(x)
  • BC: t->inf, psi->0 for + and -
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14
Q

TDSE solution method

A

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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15
Q

Inner product

A

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

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16
Q

Orthogonality

A

= delta_mn

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17
Q

Completeness

A

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

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18
Q

Transmission probability

A

Flux of stuff out / flux of stuff in

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19
Q

Reflection probability

A

Flux of stuff out backwards / flux of stuff in same

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20
Q

Finding a finite discontinuity in psi’(x)

A
  • when U(x) has an infinite jump

- integrate TISE from the inf jump left to inf jump right (ex, 0- to 0+)

21
Q

Transmission through N identical replicas with period L

A

(AN, BN) = P^N (A0, B0) where P=DM

22
Q

Eigenfunctions for SHO

A

Solutions to hermite’s DE

23
Q

Def: Hilbert space

A

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

24
Q

Hilbert space properties (7)

A

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

25
Q

Def: orthonormal basis is complete

A

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

26
Q

Def: expectation value

A

<a> = bra(psi) a ket(psi)</a>

27
Q

Def: hermitian operator

A

A* = A

=

28
Q

Formal statement for probabilities

A

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

29
Q

Def: non-degenerate

A

Iff Am = An -> |psim> = |psin>

30
Q

Def: N_n fold degenerate

A

N_n linearly independent eigvects |psi_nj> such that A|psi_nj> = An|psi_nj> for j=1, 2, …, N

31
Q

Property of hermitian operator A

A

Eigvals are real

Eigfunctions are orthogonal (non-degenerate eigvals)

32
Q

Superposition principle

A

Every physical state |f> can be represented by a generalized Fourier series

33
Q

Properties of commutator

A
  • 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
34
Q

Angular momentum in 3D

A 
Lx = yp(z) - zp(y)
Ly = zp(x) - xp(z)
Lz = xp(y) - yp(x)
35
Q

Spherical coordinates

A 
r = sqrt(x^2 + y^2 + z^2)
Theta = arccos(z/r)
Phi = arctan(y/x)
36
Q

Spherical harmonics orthonormality relation

A

= delta_m’m*delta_l’l

37
Q

Effect of magnetic field

A

Add a term to Hamiltonian: -mu*B, mu is magnetic moment

38
Q

Angular momentum: commutator relationships

A

[Li, Lj] = epsilon(ijk) ihbarL_k

39
Q

Ladder operators

A

L+- = Lx +- iLy

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

40
Q

Sz

A

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

41
Q

Sx

A

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

42
Q

Sy

A

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

43
Q

Degenerate perturbation theory

A

Hab = psia* x H1 x psib

  • compute the matrix
  • EVP: H1c = E1c where E1 = perturbation
44
Q

Parity operator properties

A
  • [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)
45
Q

What does it mean for a free particle wave to propagate without distortion?

A

Psi(x, t) = f(x-vt) where v is constant

46
Q

Debroglie wavelength relationship thing

A

k = p/hbar

47
Q

Only one bound state

A

Only one k value for E < U

48
Q

Quantized energy

A

Multiple kn

AMATH 373 (1 decks)

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