A particle of mass moves in one dimension in a real

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Wave mechanics in one dimension · Wave mechanics in one dimension

The Structure of Quantum Mechanics, Ch The wave function is not the state

Problem

A particle of mass mm moves in one dimension in a real, time-independent potential V(x)V(x). In the position representation its state obeys iℏ∂tΨ(x,t)=−ℏ22m∂x2Ψ(x,t)+V(x)Ψ(x,t).i\hbar\partial_t\Psi(x,t) = -\frac{\hbar^2}{2m}\partial_x^2\Psi(x,t)+V(x)\Psi(x,t). ((a)) Using this equation and its complex conjugate, derive the continuity equation ∂t∣Ψ(x,t)∣2+∂xj(x,t)=0,j(x,t)=ℏm Im ⁣[Ψ∗(x,t) ∂xΨ(x,t)],\partial_t|\Psi(x,t)|^2 + \partial_x j(x,t) = 0,\qquad j(x,t)=\frac{\hbar}{m}\,\mathrm{Im}\!\left[\Psi^*(x,t)\,\partial_x\Psi(x,t)\right], stating explicitly where the reality of V(x)V(x) is used. ((b)) A particle of energy EE meets a step at x=0x=0, with V(x)=0V(x)=0 for x<0x<0 and V(x)=V0V(x)=V_0 for x≥0x\ge0, where V0>EV_0>E. In the classically forbidden region the stationary state has the form Ψ(x,t)=C e−κx e−iEt/ℏ,x≥0,\Psi(x,t) = C\,e^{-\kappa x}\,e^{-iEt/\hbar},\qquad x\ge0, with CC a possibly complex constant and κ=2m(V0−E)/ℏ\kappa=\sqrt{2m(V_0-E)}/\hbar real and positive. Compute j(x,t)j(x,t) in this region. ((c)) The probability density in the forbidden region is ∣Ψ(x,t)∣2=∣C∣2e−2κx|\Psi(x,t)|^2=|C|^2e^{-2\kappa x}, nonzero and time-independent. Using the continuity equation from part (a) together with the result of part (b), explain why this is not a contradiction.

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