Suppose a particle has wavefunction ψ(x,t=0)=Aexp(−x22l2). 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 the momentum operator (p-hat), denoted as ⟨p-hat⟩, for a given wavefunction, you need to calculate the integral of the product of the wavefunction and the momentum operator over all possible values of position x.
The momentum operator p-hat is defined as -iħ(d/dx), where i is the imaginary unit (√-1), ħ is the reduced Planck's constant, and d/dx represents the derivative with respect to x.
In this case, the given wavefunction is ψ(x,t=0) = Aexp(-x^2/2l^2), where A is a constant and l^2 is a constant related to the width of the wavefunction.
To find ⟨p-hat⟩, you need to calculate the integral of the product of the wavefunction and the momentum operator over all x values.
⟨p-hat⟩ = ∫ ψ*(x,t=0)(-iħ)(d/dx)ψ(x,t=0) dx
Let's break down the steps to solve this:
Step 1: Calculate the conjugate of the wavefunction.
ψ*(x,t=0) = A*exp(-x^2/2l^2)
Step 2: Take the derivative of the wavefunction with respect to x.
(d/dx)ψ(x,t=0) = (-x/l^2) Aexp(-x^2/2l^2)
Step 3: Substitute the values into the integral expression for ⟨p-hat⟩.
⟨p-hat⟩ = ∫ A*exp(-x^2/2l^2)(-iħ)(-x/l^2) Aexp(-x^2/2l^2) dx
Step 4: Simplify the expression.
⟨p-hat⟩ = iħ∫ (x/l^2) exp(-x^2/l^2) dx
Step 5: Evaluate the integral.
This integral can be solved using integration techniques such as integration by parts or completing the square. After the integration, you will obtain the value of ⟨p-hat⟩.
It is important to note that the given wavefunction is in position space, and to find the expectation value of the momentum operator, you need to find its representation in momentum space. However, by following the steps outlined above, you can calculate the expectation value.