# Gaussian distribution integral?

• 21joanna12
In summary, the conversation discusses the integration of the quantum harmonic oscillator wave function in order to find the constant a for normalization. It is suggested to use a substitution and check for proof. The question is raised if the x^2 term should be included in the expression, and if so, whether the integral can be done in terms of elementary functions.

#### 21joanna12

when considering the quantum harmonic oscillator, you get that the wave function takes the form

$psi=ae^{-\frac{m\omega}{2\hbar}x^2}$

I have been trying to integrate $\psi ^2$ to find the constant a so that the wave function is normalised, and I know the trick with converting to polar coordinates to integrate $e^{-x^2}$, but I cannot figure out how to integrate the more complicated version above. I know that the constant should have the value $\left(\frac{m\omega}{\pi \hbar}\right)^{\frac{1}{4}}$ if the wavefunction is to be normalised, but I can't figure out how to do this?

You have
$$\int_{-\infty}^{\infty} e^{- c x^2} dx$$
Make the substitution ##\tilde{x} = \sqrt{c} x##, ##d\tilde{x} = \sqrt{c} dx##, and you get
$$\frac{1}{\sqrt{c}} \int_{-\infty}^{\infty} e^{- \tilde{x}^2} d\tilde{x}$$

Use the substitution $y=\sqrt{\frac{m\omega}{2\hbar}} x$ and check the proof here!

21joanna12 said:
when considering the quantum harmonic oscillator, you get that the wave function takes the form

$\psi=ae^{-\frac{m\omega}{2\hbar}x^2}$

I have been trying to integrate $\psi ^2$ to find the constant a so that the wave function is normalised, and I know the trick with converting to polar coordinates to integrate $e^{-x^2}$, but I cannot figure out how to integrate the more complicated version above. I know that the constant should have the value $\left(\frac{m\omega}{\pi \hbar}\right)^{\frac{1}{4}}$ if the wavefunction is to be normalised, but I can't figure out how to do this?

Do you want the $x^2$ in your expression for $\psi$? If not then the comments above should serve you well: if you do, are you really trying to integrate something
$$\propto \int_{-\infty}^\infty e^{-cx^4} \, dx$$

## 1. What is the Gaussian distribution integral?

The Gaussian distribution integral is a mathematical concept used to describe the probability distribution of a continuous random variable. It is also known as the normal distribution and is often used in statistics to model real-world phenomena.

## 2. How is the Gaussian distribution integral calculated?

The Gaussian distribution integral is calculated using a mathematical formula known as the cumulative distribution function (CDF). This formula takes into account the mean and standard deviation of the data and calculates the probability of a random variable falling within a certain range.

## 3. What is the importance of the Gaussian distribution integral in statistics?

The Gaussian distribution integral is important in statistics because it allows us to make predictions about the likelihood of a certain outcome based on a given set of data. It is also used in hypothesis testing and confidence interval calculations.

## 4. How is the concept of the Gaussian distribution integral applied in real-world situations?

The Gaussian distribution integral is applied in a wide range of real-world situations, such as in finance, engineering, and social sciences. It is commonly used to model phenomena such as stock prices, weather patterns, and human height.

## 5. Can the Gaussian distribution integral be used for non-normal distributions?

While the Gaussian distribution is often used to model real-world data, it is important to note that not all data follows a normal distribution. In cases where the data is not normally distributed, alternative statistical methods may need to be used.