Normalizing The Wave Function Of A Mass And Spring

Question

I am having some conceptual issues with understanding how a wave function is normalized using simple harmonic oscillators.

If you need to find $A$ for a simple harmonic oscillator of mass $m$ and spring constant $k$ that starts out in the state:

$$\psi(x, 0) = A(2\psi_1(x) + \psi_2(x))$$

Should this be represented as a free particle in the form

$$\psi(x, 0)=Ae^{ikx}$$

Finding $A$ means normalizing the wave function which means

$$\int_{-\infty }^{\infty } \left | \psi (x) \right |^{2}dx=1$$

But here I'm guessing I need to multiply $A$ for both $\psi_1 (x)$ and $\psi_2 (x)$ and then normalize them separately.

I am trying to understand exactly how this is suppose to be represented in 1-D simple harmonic oscillators for Schrodinger Equation. Any assistance is greatly appreciated.

Answer 1

Uh, okay, first, forget about $A e^{ikx}$. The eigenfunctions of an harmonic oscillator are well known and they can be found in any table. They are proportional to hermite polynomials.

Secondly, do not doubt: stick to the notation.

$$1=\int_{-\infty}^{+\infty} |\psi(x)|^2 dx$$

It says to do the integral of the squared modulus of the whole wavefunction.

including $A$. The value will depend on $A$, and that's how you get its value. $1=$something that depends on $A$.

So yes, you have to do

$$1=\int_{-\infty}^{+\infty} \left|\ \ A\left[\ 2\ \psi_1(x)+\psi_2(x)\ \ \right]\ \right|^2 dx $$

Now, the trick is using the complex conjugate:

$$1=\int_{-\infty}^{+\infty} \ \ A^*\left[\ 2\ \psi_1^*(x)+\psi_2^*(x)\ \ \right] \cdot A\left[\ 2\ \psi_1(x)+\psi_2(x)\ \ \right] dx $$

So you'll have $A^* A$, which is $|A|^2$ inside the integral. The rest are 4 terms, result of the distributive property. That's why we split that into 4 integrals.

Since constants can be extracted outside, the integral of $\psi_1^*\psi_2$ can be joined with the one of $\psi_2^*\psi_1$ to create twice the real part.

Anyways, that can reduce 1 integral, but there are 3 more integrals to do.

You've got two options:

1. ("Expert"): use recurrence formulae and other identities so that you end up having integrals of the form

$\int |\psi|^2 dx$, which are 1.

This is not always possible.

2. Replace the wavefunctions. You need to know the wwavefunctions (they're Hermite polynomials), and do the integrals manyally. Then, find the value of $A$.