What is the probability current density of a particle with wavefunction ψ(x,t)=exp(iℏ(px−Et))?
To calculate the probability current density of a particle with a given wavefunction, we can make use of quantum mechanics principles.
The probability current density (J) is defined as the product of the complex conjugate of the wavefunction (∗) and the gradient of the wavefunction (∇ψ):
J = ℏ/(2mi) [ψ∗∇ψ - ψ∇ψ∗]
Where:
- J is the probability current density
- ℏ is the reduced Planck's constant (ℏ = h/(2π))
- m is the mass of the particle
- i is the imaginary unit (i² = -1)
- ψ is the wavefunction
- ∗ represents the complex conjugate
- ∇ represents the gradient operator (∇ = (∂/∂x, ∂/∂y, ∂/∂z))
Let's calculate the probability current density for the given wavefunction:
ψ(x,t) = exp(iℏ(px−Et))
To find the complex conjugate of ψ, we replace i with -i:
ψ∗(x,t) = exp(-iℏ(px−Et))
Now let's calculate the gradient of the wavefunction, ∇ψ:
∇ψ = (∂/∂x, ∂/∂y, ∂/∂z)ψ
In this case, the wavefunction ψ only depends on x and t, so we only need to calculate the partial derivative with respect to x:
∂/∂xψ = (∂/∂x)(exp(iℏ(px−Et)))
Applying the chain rule, we get:
∂/∂xψ = iℏ(p)exp(iℏ(px−Et)) = iℏpψ
Now that we have ψ∗ and ∇ψ, we can substitute these values into the formula for the probability current density J:
J = ℏ/(2mi) [ψ∗∇ψ - ψ∇ψ∗]
= ℏ/(2mi) [exp(-iℏ(px−Et)) * iℏpψ - exp(iℏ(px−Et)) * (-iℏpψ)]
= ℏ/(2mi) [iℏpψ * exp(-iℏ(px−Et)) + iℏpψ * exp(iℏ(px−Et))]
Simplifying and taking out common factors:
J = ℏp/(2m) [ψ * exp(-iℏ(px−Et)) + ψ * exp(iℏ(px−Et))]
= ℏp/(2m) [2ψ * cos(ℏ(px−Et))]
Finally, we can express the wavefunction in terms of real and imaginary parts:
ψ(x,t) = A * exp(iℏ(px−Et))
= A * [cos(ℏ(px−Et)) + i * sin(ℏ(px−Et))]
Replacing ψ in the expression for J:
J = ℏp/(2m) [2A * exp(iℏ(px−Et)) * cos(ℏ(px−Et))]
The probability current density J for the given wavefunction is ℏp/(m) * A * cos(ℏ(px−Et)).