Bootcamp for Science

The Schrödinger Equation



The Schrödinger equation is given by:

(1)   \[-\frac{\hbar}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x,t)\Psi(x,t) = i\hbar\frac{\partial \Psi(x,t)}{\partial t} \]

where m is the mass of the particle, V(x,t) is the potential energy, and \hbar = h/2\pi. The function \Psi(x,t) is the wave function, representing the probability amplitude for finding the particle at position x at time t.

 

When no external force acts on the particle, the potential energy is zero, V(x,t) = 0, reducing Equation (1) to:

(2)   \[-\frac{\hbar}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2} = i\hbar\frac{\partial \Psi(x,t)}{\partial t} \]

 

From quantum mechanics, recall that E = hf and \lambda = \frac{E}{p}, where f is frequency. The wave number k is defined as:

(3)   \[k = \frac{2\pi}{\lambda}, \quad \omega = kc \]

Rearranging equations, we get:

(4)   \[f = \frac{\omega}{2\pi} = \frac{kc}{2\pi} \]

Thus, energy E is:

(5)   \[E = hf = h\frac{\omega}{2\pi} = \hbar \omega \]

And momentum p is:

(6)   \[p = \frac{h}{\lambda} = \frac{h}{2\pi} \frac{2\pi}{\lambda} = \hbar k \]

Multiplying both sides by c, we obtain:

(7)   \[E = \hbar kc = \hbar\omega = pc \]

 

Suppose \Psi(x,t) takes the form:

(8)   \[\Psi(x, t) = Ae^{i(kx-\omega t)} \]

Differentiating \Psi(x,t) with respect to x and t, we obtain:

(9)   \[\frac{\partial\Psi(x, t)}{\partial t} = -i\omega Ae^{i(kx-\omega t)} \]

(10)   \[\frac{\partial^2\Psi(x, t)}{\partial x^2} = -k^2Ae^{i(kx-\omega t)} \]

Substituting Equation (8) into Equation (2), we derive:

(11)   \[\frac{\hbar^2}{2m}(-k^2Ae^{i(kx-\omega t)}) = i\hbar (-i\omega Ae^{i(kx-\omega t)}) \]

(12)   \[\frac{\hbar^2}{2m} k^2Ae^{i(kx-\omega t)} = \hbar \omega Ae^{i(kx-\omega t)} \]

For a non-trivial solution, we obtain:

(13)   \[\frac{\hbar^2 k^2}{2m} = \hbar \omega, \quad \omega = \frac{\hbar k^2}{2m} \]

Following de Broglie, we have shown that Equations (8) and (7) hold. Assuming this applies to all particles, the total energy is:

(14)   \[E = \hbar\omega = \frac{(\hbar k)^2}{2m} = \frac{p^2}{2m} \]