Schoedinger Equation in Momentum space

[tetchou]

Please, I would like to write the time-independent schroedinger equation (describing the motion of a bound electron) in momentum space and in cylindrical coordinates.
Can you help me?
Thank you very much.
Hugues Merlain

 

[turin]

If I remember correctly, for a free particle it is:

(1/2m) |p|² ψ = (ih_bar) ∂ψ/∂t

 

[R0man]

The time-independent Schrödinger equation describes the behavior of a quantum particle in a potential field. In momentum space, it is written as:

$$\frac{\hbar^2}{2m}\left(\frac{\partial^2 \Psi}{\partial p_x^2} + \frac{\partial^2 \Psi}{\partial p_y^2} + \frac{\partial^2 \Psi}{\partial p_z^2}\right) + V\Psi = E\Psi$$

where $\Psi$ is the wave function, $m$ is the mass of the particle, $\hbar$ is the reduced Planck's constant, $p_x$, $p_y$, and $p_z$ are the momentum components, $V$ is the potential, and $E$ is the energy.

In cylindrical coordinates, the Schrödinger equation can be written as:

$$\frac{\hbar^2}{2m}\left(\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial \Psi}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 \Psi}{\partial \theta^2} + \frac{\partial^2 \Psi}{\partial z^2}\right) + V\Psi = E\Psi$$

where $r$ is the distance from the origin, $\theta$ is the angle in the xy-plane, and $z$ is the distance along the z-axis.

To solve for the wave function in momentum space, we can use the Fourier transform:

$$\Psi(p_x, p_y, p_z) = \frac{1}{(2\pi\hbar)^{3/2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \tilde{\Psi}(k_x, k_y, k_z)e^{i(k_xp_x + k_yp_y + k_zp_z)}dk_xdk_ydk_z$$

where $\tilde{\Psi}$ is the Fourier transform of $\Psi$.

Substituting this into the Schrödinger equation, we get:

$$\frac{\hbar^2}{2m}\left(-k_x^2 - k_y^2 - k_z^2\right)\tilde{\Psi}