Question 7What is the probability current density of a particle with wavefunction 𝜓(𝑥,𝑡)=exp(𝑖ℏ(𝑝𝑥−𝐸𝑡))ψ(x,t)=exp( ℏi (px−Et))?Recall that the probability current density can be computed as:𝑗=ℏ2𝑚𝑖[𝜓∗∂∂𝑥𝜓−𝜓∂∂𝑥𝜓∗]j= 2miℏ [ψ ∗ ∂x∂ ψ−ψ ∂x∂ ψ ∗ ]1 point𝑝𝐸Ep 11𝑝p𝑝𝑚mp
Question
Question 7What is the probability current density of a particle with wavefunction 𝜓(𝑥,𝑡)=exp(𝑖ℏ(𝑝𝑥−𝐸𝑡))ψ(x,t)=exp( ℏi (px−Et))?Recall that the probability current density can be computed as:𝑗=ℏ2𝑚𝑖[𝜓∗∂∂𝑥𝜓−𝜓∂∂𝑥𝜓∗]j= 2miℏ [ψ ∗ ∂x∂ ψ−ψ ∂x∂ ψ ∗ ]1 point𝑝𝐸Ep 11𝑝p𝑝𝑚mp
Solution
The given wavefunction is:
ψ(x,t) = exp(iℏ(px−Et))
The complex conjugate of the wavefunction, ψ*, is:
ψ*(x,t) = exp(-iℏ(px−Et))
The derivative of ψ with respect to x, ∂ψ/∂x, is:
∂ψ/∂x = iℏp exp(iℏ(px−Et))
The derivative of ψ* with respect to x, ∂ψ*/∂x, is:
∂ψ*/∂x = -iℏp exp(-iℏ(px−Et))
Substituting these into the formula for the probability current density, we get:
j = ℏ/2mi [ψ*∂ψ/∂x - ψ∂ψ*/∂x]
= ℏ/2mi [exp(-iℏ(px−Et)) * iℏp exp(iℏ(px−Et)) - exp(iℏ(px−Et)) * -iℏp exp(-iℏ(px−Et))]
= ℏ/2mi [iℏp - -iℏp]
= ℏ/2mi [2iℏp]
= ℏ^2p/m
So, the probability current density of the particle is ℏ^2p/m.
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