Particle in a box

Roberto Di Remigio, Luca Frediani

The particle in a box model is among the simplest, exactly solvable models in quantum mechanics. In the one-dimensional case, we assume a particle of mass $m$ to be confined into a box of length $L$. The confinement is achieved by means of a potential energy operator that is zero inside the box and infinite outside, as in the Figure below.

In practice, this means that the particle cannot escape the box: a result that we would have obtained also from classical mechanics. How does the quantum particle behave? We need to find the eigenfunctions and eigenvalues of the Hamiltonian operator, that is we have to solve the following ordinary differential equation: \begin{equation} -\frac{\hbar^2}{2m}\frac{\mathrm{d}^2}{\mathrm{d}x^2} \psi_n(x) = E_n\psi_n(x) \end{equation} with boundary conditions: \begin{equation} \begin{aligned} \psi_n(0) &= 0 \\ \psi_n(L) &= 0 \end{aligned} \end{equation} Thus acceptable solutions are of the form: \begin{equation} \psi_n(x) = \sin(\frac{n\pi x}{L}) \quad\quad \forall n \neq 0 \end{equation} with energies: \begin{equation} E_n = \frac{h^2n^2}{8mL^2} \quad\quad \forall n \neq 0 \end{equation}

Exercise 1: Normalization

The wavefunction(s) given above as solution to the particle in a box problem are not normalized: \begin{equation} \left\langle \psi_n | \psi_n \right\rangle = \int \mathrm{d}x \psi_n^*(x)\psi_n(x) = |A|^2 \neq 1 \end{equation} Find the normalization constant.

Exercise 2: Ground-state and probabilities

Given the normalized ground-state wavefunction for a particle confined to a one-dimensional box of length $L$, suppose the box is $10.0\,\,\mathrm{nm}$ long. What is the probability that the particle is:

1. Between $a = 4.95\,\,\mathrm{nm}$ and $b=5.05\,\,\mathrm{nm}$
2. Between $a = 1.95\,\,\mathrm{nm}$ and $b=2.05\,\,\mathrm{nm}$
3. Between $a = 9.90\,\,\mathrm{nm}$ and $b=10.0\,\,\mathrm{nm}$
4. In the right half of the box
5. In the central third of the box

How can we generalize to the excited states?

Exercise 3: Eigenfunctions and probability distributions

Write a Python function to plot the eigenfunctions of the particle in a box model. The function should accept the quantum number $n$, the length of the box $L$ and a NumPy array of $x$ values as arguments:

def eigenfunction1D(n, L, x):
    """ Normalized eigenfunction for the 1D particle in a box.

    n -- the quantum number
    L -- the size of the box
    x -- the NumPy array with the x values
    """

Once this function is defined, we can obtain the respective probability distribution by taking its square:

x = np.linspace(0, 10.0, 1000)
eig = eigenfunction1D(1, 10.0, x)
prob = eigenfunction1D(1, 10.0, x)**2

and plot both of them with:

plt.plot(x, eig)
plt.plot(x, prob)

Exercise 4: Normalization of linear combinations of eigenfunctions

A linear combination is defined as: \begin{equation} \Psi(x) = \sum_{i=0}^{N} c_i \psi_i(x) \end{equation} where $c_i$ are the coefficients and $\psi_i(x)$ are the eigenfunctions of the particle in a box. Is $\Psi(x)$ normalized? If not, how can we normalize it? Write a Python function that returns the normalization constant given a vector of coefficients $c_i$ in the linear combination.

def normalize(coeffs):
    """ Normalization constant for a linear combination of 1D particle in a box eigenfunctions

    coeffs -- a NumPy array with the coefficients
    """

Exercise 5: Energy of linear combinations

The expectation value of an operator $O$ on the wavefunction $\Psi$ is defined as: \begin{equation} \bar{O} = \frac{\langle\Psi|O|\Psi\rangle}{\langle\Psi|\Psi\rangle} = \langle\Psi|O|\Psi\rangle \end{equation} where the last equality holds only if $\Psi$ is normalized. The energy is the expectation value of the Hamiltonian operator: \begin{equation} \bar{H} = E = \langle\Psi|H|\Psi\rangle \end{equation}

Given $\Psi$ a linear combination of particle in a box eigenfunctions, calculate its respective energy. Hint: remember that $H\psi_i(x) = E_i\psi_i(x)$ and that the $\psi_i(x)$ are normalizd. You can also assume $\Psi$ to be already normalized.

Write a Python function to calculate the energy of such linear combinations. The function should accept the coefficients $c_i$ of the linear combination, the mass $M$ of the particle and the length $L$ of the box as arguments.

def energy(coeffs, M, L):
    """ Return energy of a linear combination of 1D particle in a box eigenfunctions.

    coeffs -- the coefficients of the linear combination
    M -- the mass of the particle
    L -- the size of the box
    """

Exercise 6: Linear combinations of eigenfunctions

We can now try to plot linear combinations of eigenfuntions and their corresponding probability densities. Define the appropriate function to do so. Your function should use your previous implementation of eigenfunction1D to achieve its purpose.