# Simple Harmonic(Quantum) Oscillator

1. Jan 1, 2005

### 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 text book 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?

Last edited: Jan 1, 2005
2. Jan 1, 2005

### 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.

3. Jan 1, 2005

### 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.