Suppose a particle has wavefunction ψ(x,t=0)=Ae^(−x^2/l^2). What is the average value (expectation value) of p, <p>, for this state at t=0?
To find the average value or expectation value of momentum (<p>) for a given wavefunction ψ(x, t=0), you need to calculate the corresponding momentum operator and then evaluate it with the wavefunction. The momentum operator is represented by the symbol "p" and is defined as:
p = -iħ(∂/∂x)
where i is the imaginary unit (√-1), ħ is the reduced Planck's constant, and (∂/∂x) denotes the partial derivative with respect to x.
Given the wavefunction ψ(x, t=0) = Ae^(-x^2/l^2), where A and l are constants, let's proceed with finding the expectation value of momentum.
Step 1: Calculate the derivative of ψ(x, t=0) with respect to x.
∂ψ(x, t=0)/∂x = (-2x/l^2)Ae^(-x^2/l^2)
Step 2: Multiply the derivative by (-iħ).
(-iħ) ∂ψ(x, t=0)/∂x = (-iħ)(-2x/l^2)Ae^(-x^2/l^2)
Step 3: Evaluate the momentum operator (-iħ∂/∂x) with the given wavefunction.
pψ(x, t=0) = (-iħ)(-2x/l^2)Ae^(-x^2/l^2)
Step 4: Find the complex conjugate of ψ(x, t=0) and multiply it by pψ(x, t=0).
ψ*(x, t=0) * pψ(x, t=0) = (Ae^(-x^2/l^2)) * (-iħ)(-2x/l^2)Ae^(-x^2/l^2)
= (-iħ)(-2x/l^2) |A|^2 e^(-2x^2/l^2)
Step 5: Integrate the expression obtained in step 4 over the entire x-axis.
∫ ψ*(x, t=0) * pψ(x, t=0) dx = (-iħ)(-2/l^2) |A|^2 ∫ x e^(-2x^2/l^2) dx
To evaluate this integral, you can use integration techniques such as substitution or integration by parts.
Once the integral is evaluated, the result will give you the average value of momentum (<p>) for the given state at t=0.