Flashcards
QM1 Final: key formulas
Cards in this deck (26)
- 1S1•S2 for two spin-1/2 particles in uncoupled basisI can also go the other way (express uncoupled operators in terms of coupled operators) using projection theorem
- 2j1•j2 in coupled basisKnow how to make a relationship like this for j1 dot J or j2 dot J as well
- 3Action of J+ or J- on state |jm>
- 4Radial part of 3D time-independent Schrodinger equation
- 51D time-independent Schrodinger equation
- 6Harmonic oscillator: Action of a+ or a- on state |n>
- 7Harmonic oscillator: x in terms of a+ and a-
- 8Harmonic oscillator: px in terms of a+ and a-
- 9Schrodinger equation (governs time-evolution of states in Schrodinger picture)
- 10Time evolution operator
- 11Equation for time-evolution of operators in Heisenberg picture
- 12Commutator of x and px
- 13WKB energy quantization approximation
- 14Wigner-Eckart theorem
- 15The probability of a dipole transition is proportional to...?
- 16Spherical tensor operator components of r in terms of cartesian components of r
- 17J+ and J- in terms of Jx and Jy
- 18What statistics do different types of particles obey?Bosons (spin 0, 1, 2, etc) must be in states that are overall symmetric under exchange.
Fermions (spin 1/2, 3/2, 5/2, etc) must be in states that are overall antisymmetric under exchange. - 19Symmetry of the J states for j_1 = j_2J = 2 j1 is symmetric
J = 2 j1 - 1 is antisymmetric
J = 2 j1 - 2 is symmetric
etc - 20Eigenvalues of 2 state system
- 212 state system: how to evolve a state over timeMethod 1: Guess that the elements of the time-evolved state have the form Ae^-iω1t and Be^-iω2t. Substitute into Schrodinger equation and solve for unknowns.
Method 2: Express state as linear combination of energy eigenstates & apply time-evolution operator.
Method 3: Express H as hbar ω times a normalized linear combination of Pauli matrices σ•n, & express time-evolution operator as
cos(ωt)1 - i (σ•n) sin(ωt). Apply to initial state using matrix multiplication. Note that hbar ω should come out to be sqrt(epsilon^2 + |delta|^2) for properly normalized sigma. - 22Eigenvectors of a 2 state systemEdit: Delta should be |Delta|exp(-i phi)
(note the new minus sign)
- 23Pauli matrices
- 24Pauli matrix property: exp(+/- iωt(σ•n)) = ?
- 25First few spherical harmonics (l=0, l=1)
- 26Three Spherical Harmonic Formula
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- Physics Flashcards on QM1 Final: key formulas, created by Yorick Andeweg on 15/12/2021.
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- Yorick Andeweg
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