# Inner product of two wave functions is constant in time

Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 1.16.

The fact that the normalization of the wave function is constant over time is actually a special case of a more general theorem, which is

$\displaystyle \frac{d}{dt}\int_{-\infty}^{\infty}\Psi_{1}^*\Psi_{2}dx=0 \ \ \ \ \ (1)$

for any two normalizable solutions to the Schrödinger equation (with the same potential). The proof of this follows a similar derivation to that in section 1.4 of Griffiths’s book.

The derivative in the integrand is (where we’re using a subscript ${t}$ or ${x}$ to denote a derivative with respect that variable):

 $\displaystyle \frac{\partial}{\partial t}\left(\Psi_{1}^*\Psi_{2}\right)$ $\displaystyle =$ $\displaystyle \Psi_{1t}^*\Psi_{2}+\Psi_{1}^*\Psi_{2t} \ \ \ \ \ (2)$

From the Schrödinger equation

 $\displaystyle \Psi_{2t}$ $\displaystyle =$ $\displaystyle i\frac{\hbar}{2m}\Psi_{2xx}-\frac{i}{\hbar}V\Psi_{2}\ \ \ \ \ (3)$ $\displaystyle \Psi_{1t}^*$ $\displaystyle =$ $\displaystyle -i\frac{\hbar}{2m}\Psi_{1xx}^*+\frac{i}{\hbar}V\Psi_{1}^*\ \ \ \ \ (4)$ $\displaystyle \Psi_{1t}^*\Psi_{2}+\Psi_{1}^*\Psi_{2t}$ $\displaystyle =$ $\displaystyle i\frac{\hbar}{2m}\left(-\Psi_{1xx}^*\Psi_{2}+\Psi_{2xx}\Psi_{1}^*\right)+\frac{i}{\hbar}V\left(\Psi_{1}^*\Psi_{2}-\Psi_{1}^*\Psi_{2}\right)\ \ \ \ \ (5)$ $\displaystyle$ $\displaystyle =$ $\displaystyle i\frac{\hbar}{2m}\frac{\partial}{\partial x}\left(\Psi_{2x}\Psi_{1}^*-\Psi_{1x}^*\Psi_{2}\right) \ \ \ \ \ (6)$

Inserting this into 1 and integrating gives zero because all wave functions go to zero at infinity. [Of course, the theorem doesn’t hold if ${\Psi_{1}}$ and ${\Psi_{2}}$ are solutions for different potentials, because in that case the potential term wouldn’t cancel out in 5.]

## One thought on “Inner product of two wave functions is constant in time”

1. Kamila

Hello Sir, I really appreciate your work, but could you please tell something about the physical meaning of this theorem? I cannot find it anywhere…