# Quantum Physics I Tough Normalization Integral Help

• Kvm90
In summary, the conversation was about normalizing a given wave function and finding the answer using the Gaussian Integral. The answer is given as [sqrt(2)/pi^(1/4)][km/h^2]^(3/8).
Kvm90
I am pretty sure that I'm doing this right but the integral for normalization seems impossible. Here is the question:

Normalize this wave function.
psi(x,t)=Axe^(-sqrt(km)/2h)*x^2))*e^(-i(3/2)(sqrt(k/m)*t)
for -infinity<x<+infinity where k and A are constants and m is given.

I used int(psi(x,t)psi*(x,t)dx)=1 and have arrived at the following integral, which I (nor online integrators) can't clearly do:

int(A^2*x^2*e^[(-sprt(km)/h)*x^2]=1

According to the back of the book, the answer is [sqrt(2)/pi^(1/4)][km/h^2]^(3/8). Any help would be appreciated!

This is an example of a Gaussian Integral, which is given by the following:

$$\int_{-\infty}^{\infty} e^{-\alpha x^2} dx = \sqrt{\frac{\pi}{\alpha}}$$

In your case if you let $\alpha = \left ( \frac{km}{h} \right )^{1/2}$, I got the following when I used Wolfram Alpha:

$$\int_{- \infty}^{\infty} x^2 e^{-ax^2} dx = \frac{\sqrt{\pi}}{2a^{3/2}}=\frac{1}{A^2}$$

Now just solve for A

## 1. What is a normalization integral in quantum physics?

A normalization integral in quantum physics is a mathematical tool used to determine the probability of a particle being in a certain state. It is necessary because the wavefunction, which describes the state of a particle, must be normalized to 1 in order to accurately represent the probability of finding the particle in a specific location.

## 2. Why is normalization important in quantum physics?

Normalization is important in quantum physics because it ensures that the total probability of finding a particle in any location is equal to 1. This is necessary for the wavefunction to accurately represent the behavior of particles at the quantum level.

## 3. How do you calculate a normalization integral?

To calculate a normalization integral, you first square the wavefunction, which gives you the probability density. Then, you integrate the probability density over all possible values of the variable (such as position or momentum). Finally, you take the square root of the result to get the normalized wavefunction.

## 4. What is the role of the normalization constant in a normalization integral?

The normalization constant is a numerical value that is multiplied by the wavefunction after the integration is performed. It is used to ensure that the total probability of finding the particle in any location is equal to 1. Without this constant, the wavefunction would not accurately represent the probability of finding the particle in different locations.

## 5. Can you give an example of a normalization integral in quantum physics?

One example of a normalization integral is the calculation of the wavefunction for a free particle. In this case, the wavefunction is a plane wave, and the normalization integral involves integrating the square of the wavefunction over all possible values of momentum. The resulting normalized wavefunction is a constant value, indicating that the particle has an equal probability of being found in any location.

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