- #1
Cogswell
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Homework Statement
Calculate ## \dfrac{d <p>}{dt} ##
Answer: ## \left< -\dfrac{\partial V}{\partial x} \right> ##
Homework Equations
Schrodinger equation: ## i \hbar \dfrac{\partial \Psi}{\partial t} = -\dfrac{\hbar ^2}{2m} \frac{\partial ^2 \Psi}{\partial x^2} + V \Psi ##
The Attempt at a Solution
Here's what I did:
## \displaystyle \dfrac{\partial}{\partial t} \int^{\infty}_{- \infty} \Psi ^* \left( \dfrac{\hbar}{i} \dfrac{\partial}{\partial x} \right) \Psi dx ##
## \displaystyle \dfrac{\hbar}{i} \int^{\infty}_{- \infty} \dfrac{\partial}{\partial t} \Psi ^* \dfrac{\partial \Psi}{\partial x} dx ##
## \displaystyle \dfrac{\hbar}{i} \int^{\infty}_{- \infty} \left[ \dfrac{\partial \Psi ^*}{\partial t} \dfrac{\partial \Psi}{\partial x} + \Psi^* \dfrac{\partial}{\partial t} \dfrac{\partial \Psi}{\partial x} \right] dx ## (Differentiation by Product rule)From the Schrodinger equation we get that: ## \dfrac{\partial \Psi}{\partial t} = \dfrac{i \hbar}{2m} \dfrac{\partial ^2 \Psi}{\partial x^2} - \dfrac{i}{\hbar} V \Psi ##
And it's conjugate: ## \dfrac{\partial \Psi ^*}{\partial t} = -\dfrac{i \hbar}{2m} \dfrac{\partial ^2 \Psi^*}{\partial x^2} + \dfrac{i}{\hbar} V \Psi^* ##
Putting those into my integral I get:
## \displaystyle \dfrac{\hbar}{i} \int^{\infty}_{- \infty} \left[ \left( -\dfrac{i \hbar}{2m} \frac{\partial ^2 \Psi^*}{\partial x^2} + \dfrac{i}{\hbar} V \Psi^* \right) \dfrac{\partial \Psi}{\partial x} + \Psi^* \dfrac{\partial}{\partial x} \left( \dfrac{i \hbar}{2m} \frac{\partial ^2 \Psi}{\partial x^2} - \dfrac{i}{\hbar} V \Psi \right) \right] dx ##
Expanding out everything:
## \displaystyle \dfrac{\hbar}{i} \int^{\infty}_{- \infty} \left[ -\dfrac{i \hbar}{2m} \frac{\partial ^2 \Psi^*}{\partial x^2} \dfrac{\partial \Psi}{\partial x} + \dfrac{i}{\hbar} V \Psi^* \dfrac{\partial \Psi}{\partial x} + \dfrac{i \hbar}{2m} \frac{\partial ^3 \Psi}{\partial x^3} \Psi^* - \dfrac{i}{\hbar} \Psi^* \dfrac{\partial}{\partial x} (V \Psi) \right] dx ##
## \displaystyle \dfrac{\hbar}{i} \int^{\infty}_{- \infty} \left[ -\dfrac{i \hbar}{2m} \frac{\partial ^2 \Psi^*}{\partial x^2} \dfrac{\partial \Psi}{\partial x} + \dfrac{i}{\hbar} V \Psi^* \dfrac{\partial \Psi}{\partial x} + \dfrac{i \hbar}{2m} \frac{\partial ^3 \Psi}{\partial x^3} \Psi^* - \dfrac{i}{\hbar} \dfrac{\partial V}{\partial x} \Psi \Psi * - \dfrac{i}{\hbar} \dfrac{\partial \Psi}{\partial x} V \Psi ^* \right] dx #### \displaystyle \dfrac{\hbar}{i} \int^{\infty}_{- \infty} \left[ \underbrace{-\dfrac{i \hbar}{2m} \frac{\partial ^2 \Psi^*}{\partial x^2} \dfrac{\partial \Psi}{\partial x}}_1 + \underbrace{\dfrac{i \hbar}{2m} \frac{\partial ^3 \Psi}{\partial x^3} \Psi^*}_2 - \underbrace{\dfrac{i}{\hbar} \dfrac{\partial V}{\partial x} \Psi \Psi *}_3 \right] dx ##
I'm stuck at this point. I'm presuming there's a way to cancel out each of the integrals? I know the last integral is the one I want but I do not know how to cancel out the first 2.