Fermi · Concept Library · Courses · Books · Problems · Simulations · Math Wiki · High School Fermi › Problems Franz Schwabl Quantum Mechanics 4th Edit · Problem 7.1
Operators, Matrices, State Vectors
Physics — Quantum Mechanics · continuity-equation-electromagnetic
Problem Let the Hamiltonian be
H = 1 2 m ( p − e c A ( x , t ) ) 2 + e Φ ( x , t ) . H = \frac{1}{2m}\left(\boldsymbol{p} - \frac{e}{c}\boldsymbol{A}(\boldsymbol{x},t)\right)^2 + e\Phi(\boldsymbol{x},t) \;. H = 2 m 1 ( p − c e A ( x , t ) ) 2 + e Φ ( x , t ) . ( ∂ / ∂ t ) ψ ∗ ψ + ∇ ⋅ j = 0 (\partial/\partial t)\psi^*\psi + \boldsymbol{\nabla}\cdot\boldsymbol{j} = 0 ( ∂ / ∂ t ) ψ ∗ ψ + ∇ ⋅ j = 0 j ≡ ℏ 2 m i [ ψ ∗ ∇ ψ − ( ∇ ψ ∗ ) ψ − 2 i e ℏ c A ( x , t ) ψ ∗ ψ ] ≡ 1 2 m ( ψ ∗ ( ℏ i ∇ − e c A ( x , t ) ) ψ + c.c. ) . \boldsymbol{j} \equiv \frac{\hbar}{2mi}\left[\psi^*\boldsymbol{\nabla}\psi - (\boldsymbol{\nabla}\psi^*)\psi - \frac{2ie}{\hbar c}\boldsymbol{A}(\boldsymbol{x},t)\psi^*\psi\right] \equiv \frac{1}{2m}\left(\psi^*\left(\frac{\hbar}{\mathrm{i}}\boldsymbol{\nabla} - \frac{e}{c}\boldsymbol{A}(\boldsymbol{x},t)\right)\psi + \text{c.c.}\right)\;. j ≡ 2 mi ℏ [ ψ ∗ ∇ ψ − ( ∇ ψ ∗ ) ψ − ℏ c 2 i e A ( x , t ) ψ ∗ ψ ] ≡ 2 m 1 ( ψ ∗ ( i ℏ ∇ − c e A ( x , t ) ) ψ + c.c. ) . Browse the problem library