Simple Harmonic(Quantum) Oscillator

[Hyperreality]

Using the normalization condition, show that the constant A has the value (\frac{m\omega_0}{\hbar\pi})^{1/4}.

I know from text the textbook that

\psi(x)=Ae^{-ax^2}

where A is the amplitdue and a=\frac{\sqrt{km}}{2\hbar}

Here is my working:

Because the motion of the particle is confined to -A to +A, so the probability of finding the particle in the interval of -A to +A must be 1. Therefore the normalization condition is

\int_{-A}^{A}|\psi(x)^2| dx = 1
A^2\int_{-A}^{A} e^{-2ax^2} dx = 1

Here's where I'm stuck, this equation cannot be solved via integration techniques, it can only be solved using by numerical methods. I only know the "trapezium rule" and the "Simpson's Rule", I tried both of methods but nothing came up. Does this problem require some other numerical methods or is my normalization condition incorrect?

 

[Dr Transport]

Normalization is over the entire region, i.e. -\infty to +\infty, the integral is then solved very easily.

A^2\int_{-\infty}^{\infty} e^{-2ax^2} dx = 1 for a = 1 do a change of variables to get the correct answer, hence the normalization factor.

 

[Tide]

Have you stated the problem correctly? It appears your limits of integration are the same as the amplitude which doesn't make sense because they have different units.