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The Quantum Harmonic Oscillator in the ground state is specified by the following Gaussian PDF in two dimensions:

$$p(x,y)= \frac{M \omega_x}{\pi h}\sqrt{ \frac{\omega_y}{\omega_x}} e^{-\frac{M}{h}(\omega_xx^2+\omega_yy^2)}$$

where M = mass of particle, h = plank's constant, $\omega_x=\frac{h}{2M\sigma_x^2}$ and $\omega_y=\frac{h}{2M\sigma_y^2}$ are related to the variances $\sigma_x$ and $\sigma_y$in the x and y direction respectively.

How do I draw a 1000 samples from a normal distribution like this in MATLAB?

I do not think I can simply use randn(1000,1) because I need to somehow take into consideration the constants $M,h,\pi, \omega_x, \omega_y$ when drawing samples. Is there way to accomplish this in MATLAB assuming all these constants are predefined?

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Gaussian PDF is separable in its variables:

$$p(x,y)=p_x(x)p_y(y).$$

This means that your $x$ and $y$ random variables are independent and can be generated from their respective PDFs $p_x$ and $p_y$. After you get expressions for (normalized!) $p_x$ and $p_y$ you can simply use inverse transform sampling to generate $x$ and $y$ respectively.

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  • $\begingroup$ @Ruslan- Thanks for your insight. I just factored out $p(x,y)$ like you said above and then plugged in $\omega_x$ into $p_x(x)$ and $\omega_y$ into $p_y(y)$ and wind up getting the normal distribution with zero mean and variances $\sigma_x$ and $\sigma_y$ respectively. Do you know what the typical values of the variance of a QHO is? Does it make sense to set the variance in the x direction to 1 and the variance in the y direction to 10? is that behavior reflective of reality? $\endgroup$ – user1068636 May 19 at 14:28
  • $\begingroup$ Harmonic oscillator is a very general model. What typical values are totally depends on your application. $\endgroup$ – Ruslan May 19 at 15:01

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