physref

Fundamental equations of Quantum Mechanics (WIP)

  1. Schrödinger representation
  2. Heisenberg representation

Schrödinger representation

In the Schrödinger picture, state vectors evolve in time but operators are constant.

Momentum operator
p^=−iℏ∇\hat{p}=-i\hbar\nabla
(1)
Hamiltonian operator
H^=−ℏ22m∇2+V(r)\hat{H}=-\dfrac{\hbar^2}{2m}\nabla^2+V(\mathbf{r})
(2)
Time-independent Schrödinger equation (TISE)
H^ψn(r)=Enψn(r)\hat{H}\psi_n(\mathbf{r}) = E_n\psi_n(\mathbf{r})
(3)
Time-dependent Schrödinger equation (TDSE)
H^Ψ(r,t)=iℏdΨ(r,t)dt\hat{H}\Psi(\mathbf{r},t) = i\hbar\dfrac{d\Psi(\mathbf{r},t)}{dt}
(4)
  • General solution to the TDSE
    Ψ(r,t)=∑n=1∞cnψn(r)e−iEnt/ℏ\Psi(\mathbf{r},t)=\sum\limits_{n=1}^\infty c_n\psi_n(\mathbf{r})e^{-i{E_n}t/\hbar}
    (5)
  • Probability of measuring eigenvalue $E_n$
    ∣cn∣2=∣⟨fn∣Ψ⟩∣2|c_n|^2 = |⟨f_n|\Psi⟩|^2
    (6)
Expectation value of an operator
⟨Q^⟩=⟨Ψ∣Q^∣Ψ⟩=∫Ψ∗Q^Ψdx=∑En∣cn∣2⟨\hat{Q}⟩ = ⟨\Psi|\hat{Q}|\Psi⟩ = \int{\Psi^* \hat{Q} \Psi dx} = \sum{{E_n}{|c_n|^2}}
(7)
Time dependence of expectation value
d⟨Q^⟩dt=iℏ⟨[H^,Q^]⟩+⟨∂Q^∂t⟩\dfrac{d⟨\hat{Q}⟩}{dt} = \dfrac{i}{\hbar} ⟨[\hat{H}, \hat{Q}]⟩ + \bigg⟨ \dfrac{\partial \hat{Q}}{\partial t} \bigg⟩
(8)
Variance
σQ2=⟨(Q^−⟨Q^⟩)2⟩=⟨Q^2⟩−⟨Q^⟩2\sigma_Q^2 = ⟨(\hat{Q}-⟨\hat{Q}⟩)^2⟩ = ⟨\hat{Q}^2⟩ - ⟨\hat{Q}⟩^2
(9)
Standard deviation
σQ=σQ2\sigma_Q = \sqrt{\sigma_Q^2}
(10)
General uncertainty principle
σAσB⩾∣12i⟨[A^,B^]⟩∣\sigma_A \sigma_B \geqslant |\tfrac{1}{2i}⟨[\hat{A}, \hat{B}]⟩|
(11)
  • Heisenberg uncertainty principle
    σxσp⩾ℏ2\sigma_x \sigma_p \geqslant \tfrac{\hbar}{2}
    (12)
  • Energy-time uncertainty principle
    ΔEΔt⩾ℏ2\Delta E \Delta t \geqslant \tfrac{\hbar}{2}
    (13)

Heisenberg representation

In the Heisenberg picture, operators evolve in time but state vectors are constant.

Density operator
ρ^(t)=∑kpkρk(t)\hat{\rho}(t) = \sum\limits_k {{p_k}{\rho _k}(t)}
(14)

Trρ^=1{\text{Tr}} \hat{\rho} = 1

  • ρk(t)=∣ψk(t)⟩⟨ψk(t)∣\rho_k(t) = |\psi_k(t)⟩⟨\psi_k(t)|
    (15)
  • pk is probability of being in state ∣ψk(t)⟩p_k \text{ is probability of being in state } |\psi_k(t)⟩
    (16)
Ensemble average
⟨A^⟩(t)=Tr{ρ^(t)A^}⟨\hat{A}⟩(t) = {\text{Tr}}\{{\hat{\rho}(t)\hat{A}}\}
(17)
Time-evolution of density operator
iℏdρ^(t)dt=[H^(t),ρ^(t)]i\hbar \dfrac{d\hat{\rho}(t)}{dt} = [{\hat{H}(t), \hat{\rho}(t)}]
(18)