In quantum mechanics the particle in a box, i.e. a particle trapped in an infinite potential well, demonstrates one of the
key differences between classical and quantum mechanics. The scenario consists of a particle that is free to move within
a certain region, in this case a two-dimensional one, which is the "box". The potential within the box being \(V=0\)
with the potential outside being \(V=\infty\) which traps the particle within the region with zero potential. The dimensions of the box
are denoted as \( a \times b \) as shown in te diagram below:
Using either the time-dependent Schrödinger equation,
\[ i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^{2}}{2m^2} \nabla^{2}\psi + V(x)\psi, \]
or the time-independent Schrödinger equation,
\[ E\psi = -\frac{\hbar^2}{2m}\nabla^2 \psi + V(x)\psi ,\]
we can find the wave function of the particle. The wave function \( \psi(x,y) \) of a particle describes its quantum state.
Using the time-independent Schrödinger equation the wave function at \(t=0\) can be found which I'll denote as
\[ \psi_{n_x,n_y}(x,y,0) = \frac{2}{\sqrt{ab}} \sin{(\frac{n_x \pi x}{a})} \sin{(\frac{n_y \pi y}{b})} .\]
Since this is the equation for a plane wave, since we are considering a free particle, we
know that the time dependence of a plane wave can be represented as,
\[ \psi_{n_x,n_y}(x,y,t) = \psi(x,y,0)e^{-iE\frac{t}{\hbar}} ,\]
and using Euler's identity we can write this as
\[ \psi_{n_x,n_y}(x,y,t) = \frac{2}{\sqrt{ab}} \sin{(\frac{n_x \pi x}{a})} \sin{(\frac{n_y \pi y}{b})} \cos{(E\frac{t}{\hbar})}
- i\frac{2}{\sqrt{ab}} \sin{(\frac{n_x \pi x}{a})} \sin{(\frac{n_y \pi y}{b})} \sin{(E\frac{t}{\hbar})} .\]
Notice that this function is complex valued.
The formula for the allowed discrete energies is
\[ E_{n_x,n_y} = \frac{\hbar^2 \pi^2}{2m}(\frac{n_x^2}{a^2} + \frac{n_y^2}{b^2}) .\]
To get the probability that the particle is at a certain position \( (x,y) \)
at a certain time \( t \) one has to take the modulus of the wave function and square it. Thus the probability
density function is defined as
\[ \rho(x,y,t) = |\psi_{n_x,n_y}(x,y,t)|^2 .\]
The modulus is taken because the wave function can be both negative & complex valued while probabilities
cannot be.
The solution for the corresponding values of \( n_x , n_y , a, b\) are shown below. The real and
imaginary part of the wave function are plotted on separate plots:
As can be seen from the plots of the time evolution the wave function changes as \(t\)
does. However the probability density function \(\rho(x,y,t)\) does not change with time. This
makes sense mathematically if we remember that \( |z_1 z_2| = |z_1||z_2|\) so we have
\[|\psi_{n_x,n_y}(x,y,t)|^2 = |\psi_{n_x,n_y}(x,y,0)e^{-iE\frac{t}{\hbar}}|^2 = |\psi_{n_x,n_y}(x,y,0)|^2 |e^{-iE\frac{t}{\hbar}}|^2 . \]
Since \(|e^{ix}| = 1 \) this reduces to
\[ |\psi_{n_x,n_y}(x,y,t)|^2 = |\psi_{n_x,n_y}(x,y,0)|^2 .\]
Which has no time dependence. This is a consequence of the fact that the wave function of
the particle is not a superposition of two or more exact energy state solutions.
However if the particle begins in a superposition of two or more different exact energy state solutions
such as \( E_{1,2} \) and \( E_{1,3} \) then the initial wave function becomes,
\[ \psi(x,y,0) = \psi_{1,2}(x,y,0) + \psi_{1,3}(x,y,0) ,\] and the time-dependent wave function is
\[ \psi(x,y,t) = \psi_{1,2}(x,y,0)e^{-iE_{1,2}\frac{t}{\hbar}} + \psi_{1,3}(x,y,0)e^{-iE_{1,3}\frac{t}{\hbar}} .\]
And the probability density function becomes,
\[ |\psi(x,y,t)|^2 = |\psi_{1,2}(x,y,0)e^{-iE_{1,2}\frac{t}{\hbar}} + \psi_{1,3}(x,y,0)e^{-iE_{1,3}\frac{t}{\hbar}}|^2 ,\]
which can be simplified through algebra to be
\[ \rho(x,y,t) = \psi_{1,2}(x,y,0)^2 + \psi_{1,3}(x,y,0)^2 + 2\psi_{1,2}(x,y,0)\psi_{1,3}(x,y,0)\cos{(E_{1,2}\frac{t}{\hbar} - E_{1,3}\frac{t}{\hbar})} .\]
Below is the simulator that shows the new behavior:
Notice that when the particles wave function is a
superposition of two different exact energy solutions the probability distribution changes with time and thus
we have the physics of motion in quantum mechanics. However since the wave function is a superposition
of two exact energy states there is now some uncertainty \( \Delta E\) in the energy of the particle even for
macroscopic objects like a ball. However this uncertainty is negligible at that scale. You may also notice that
sometimes when you have 2 sets of different quantum numbers \( (n_{x_1}, n_{y_1}) \) and \( (n_{x_2}, n_{y_2}) \)
the probability distribution is static. This is because of something called degeneracy which occurs when
2 sets of combinatorially different quantum number have the same energy \(E\), and example being
\( (n_{x_1}=1, n_{y_1}=3) \) and \( (n_{x_2}=3, n_{y_2}=1) \).
