Schrodinger equation, normalizing

[SoggyBottoms]

Homework Statement

Consider the infinite well, a particle with mass m in the potential

$$V(x) = \begin{cases} 0, & 0 < x < a,\\ \infty, & \text{otherwise,} \end{cases},$$

At t = 0 the particle is in the state:

$$\Psi(x,0) = B \left[\sin{\left(\frac{l \pi}{a}x\right)} + b\sin{\left(\frac{2l \pi}{a}x\right)}\right]$$
with b a real number and l a whole number. Use normalization to show how B depends on b.

$$1 = B^2 \int_0^a \left[\sin{\left(\frac{l \pi}{a}x\right)} + b\sin{\left(\frac{2l \pi}{a}x\right)}\right]^2 dx \\ = B^2 \left(\frac{1}{2} \int_0^a (1 - \cos{\left(\frac{2 l \pi}{a}x\right)})dx + \frac{b^2}{2} \int_0^a (1 - \cos{\left(\frac{4 l \pi}{a}x\right)})dx + b\int_0^a \cos{\left(\frac{l \pi}{a}x\right)}dx + b\int_0^a \cos{\left(\frac{3 l \pi}{a}x\right)}dx\right)\\ = B^2(\frac{a}{2} + \frac{b^2a}{2})$$

$B = \sqrt{\frac{2}{a(1 + b^2)}}$

Did I do this correctly?

 

[ehild]

It looks correct.

ehild

 

[SoggyBottoms]

Thanks.

 

[SoggyBottoms]

Now I have to calculate the expectation value of the energy. $E = \frac{p^2}{2m}$, so $\langle E \rangle = \langle \frac{p^2}{2m} \rangle$ and with $\langle p \rangle = \int_{-\infty}^{\infty} \Psi^{\ast} \frac{\hbar}{i} \frac{\partial}{\partial x} \Psi dx$ we get:

$$\langle E \rangle = \frac{-\hbar^2}{2m} \frac{2}{a(1 + b^2)} \int_0^a \Psi^{\ast} \frac{\partial^2}{\partial x^2} \Psi dx$$

Is the integrand simply equal to $\frac{d^2}{dx^2} \left[\sin{\left(\frac{l \pi}{a}x\right)} + b\sin{\left(\frac{2l \pi}{a}x\right)}\right]^2$? Because that becomes a really long calculation, so I was wondering if there is another way...

 

[asteriatic]

Yes, your calculations and final expression for B are correct. The normalization condition ensures that the wave function is properly normalized, meaning that the total probability of finding the particle within the infinite well is equal to 1. This is an important concept in quantum mechanics and it is necessary to ensure that the solutions to the Schrodinger equation accurately describe the physical system. Good job!