Find the expectation value of the linear momentum

[s8wardwe]

Homework Statement

For a given wave function Psi(x,t)=Aexp^-(x/a)^2*exp^-iwt*sin(kx) find the expectation value of the linear momentum.

Homework Equations

<p>=integral(-inf,inf) psi* p^ psi dx
p^=-ih(bar) d/dx
sin x = (exp ix - exp -ix)/2i
cos x = (exp ix + exp -ix)/2

The Attempt at a Solution

I understand the technique of sandwiching the operator between the wave function and it's complex conjugate. Then the integral is a mess of sines cosines and exponentials. I was wondering if anyone had any advice to simplify the expression or to solve this type of infinite integral. Your suggestions would be very helpful. Thanks in advance.

 

[buffordboy23]

Look at the individual integrands and determine if they are odd or even functions. If the integrand is odd and if the integration interval is symmetric with respect to some origin (i.e. negative infinity to positive infinity or [-a,a]), you can exploit the fact that integrand integrates to zero. For even functions with symmetric intervals, you can multiply the integral by 2 and run the integral from 0 to your upper boundary value.

$$\int^{a}_{-a}f_{odd}\left(x\right) dx = 0$$

$$\int^{a}_{-a}f_{even}\left(x\right) dx = 2 \int^{a}_{0}f_{even}\left(x\right) dx$$

The formula for even functions is useful for exponential terms, like e^(x), when the integration interval runs from negative to positive infinity.