Quantum Dynamics: Analytical & Computational Solutions in Python
Master time evolution in quantum mechanics. Learn to solve the harmonic oscillator and double-well tunneling using both analytical math and computational Python.
Problem 1: The Truncated Parabola in a Harmonic Oscillator
Problem Statement
Consider a particle of mass \(m\) subject to a one-dimensional quantum harmonic oscillator potential: \[V(x)=\frac{1}{2}m\omega^2x^2\]
At time \(t=0\), the particle is prepared in the initial state: \[\psi(x,0)=\begin{cases}N(A-Bx^2)&\text{for }x^2\le\frac{A}{B}\\0&\text{for }x^2>\frac{A}{B}\end{cases}\]
where \(A\) and \(B\) are positive real constants, and \(N\) is the normalization constant.
Tasks
Determine the normalization constant \(N\).
Calculate the probability of finding the particle in the energy eigenstates \(E_n\).
Write down the expression for the time-evolved wave function \(\psi(x,t)\).
Solution & Dynamics
Analytical Solution
1. Finding the Normalization Constant \(N\)
To normalize the wavefunction \(\psi(x,0)\), we require the integral of its probability density over all space to equal \(1\): \[\int_{-\infty}^{\infty} |\psi(x,0)|^2 dx = 1\]
Since the wavefunction is zero for \(x^2 > A/B\), the integration limits are restricted to \(\pm \sqrt{A/B}\): \[\int_{-\sqrt{A/B}}^{\sqrt{A/B}} N^2 (A - Bx^2)^2 dx = 1\]
We expand the integrand and use the symmetry (even function) to double the integral from \(0\) to \(\sqrt{A/B}\): \[2N^2 \int_{0}^{\sqrt{A/B}} (A^2 - 2ABx^2 + B^2x^4) dx = 1\]
Solving for \(N\) (choosing the positive root): \[\boxed{N = \frac{\sqrt{15} B^{1/4}}{4 A^{5/4}}}\]
2. Overlap Coefficients & Probabilities
The probability \(P_n\) of measuring the energy eigenvalue \(E_n = (n + \frac{1}{2})\hbar\omega\) is given by \(P_n = |c_n|^2\), where \(c_n\) is the overlap integral of the initial state with the \(n\)-th eigenstate: \[c_n = \int_{-\infty}^{\infty} \psi_n(x) \psi(x,0) dx\]
The normalized eigenstates of the 1D quantum harmonic oscillator are: \[\psi_n(x) = \left(\frac{\alpha^2}{\pi}\right)^{1/4} \frac{1}{\sqrt{2^n n!}} H_n(\alpha x) e^{-\frac{1}{2}\alpha^2 x^2}\] where \(\alpha = \sqrt{m\omega/\hbar}\).
Parity (Symmetry) Argument: The initial wavefunction \(\psi(x,0)\) is symmetric about the origin (even parity). The \(n\)-th eigenfunction \(\psi_n(x)\) has the parity of \((-1)^n\).
For odd \(n\), \(\psi_n(x)\) is an odd function. The product of an even and an odd function is odd, making the integral over the symmetric interval \([-x_0, x_0]\) vanish: \[\boxed{c_n = 0 \quad \text{for odd } n \implies P_n = 0}\]
For even \(n\), \(\psi_n(x)\) is an even function, and the overlap is non-zero: \[c_n = 2 N \left(\frac{\alpha^2}{\pi}\right)^{1/4} \frac{1}{\sqrt{2^n n!}} \int_{0}^{\sqrt{A/B}} (A - Bx^2) H_n(\alpha x) e^{-\frac{1}{2}\alpha^2 x^2} dx\] Evaluating this integral analytically for general even \(n\) yields complex expressions containing special functions. Thus, we compute it numerically using Simpson’s integration rule.
3. Time Evolution
The time-evolved wavefunction \(\psi(x,t)\) is constructed using the expansion: \[\psi(x,t) = \sum_{n=0}^{\infty} c_n \psi_n(x) e^{-i E_n t / \hbar}\]
Substituting \(c_n = 0\) for odd \(n\), this simplifies to a summation over even eigenstates: \[\boxed{\psi(x,t) = \sum_{k=0}^{\infty} c_{2k} \psi_{2k}(x) e^{-i E_{2k} t / \hbar}}\] where \(E_{2k} = \left(2k + \frac{1}{2}\right)\hbar\omega\).
Computational Solution
Below is the Python implementation that solves the normalization, computes the coefficients, plots the 2-panel physical dashboard, and renders the live interactive time-evolution of the wavefunction. We hardcode physical parameters representing an electron in an optical harmonic potential:
Left: Potential well V(u) along with the initial wavefunction (shaded gray) and time-evolved wavefunction (colored lines) at t = T/4. Right: Energy eigenstate probability spectrum.
Live Time Evolution Animation
The code below compiles a continuous time-evolution of the wavefunction over two periods (\(t \in [0, 2T]\)). The engine will render it inline as an interactive JavaScript player:
A particle of mass \(m\) in a harmonic oscillator potential \(V(x)=\frac{1}{2}m\omega^2x^2\) is initially prepared in a Gaussian wave packet given by: \[\psi(x,0)=Ne^{-\frac{(x-x_0)^2}{2a^2}}\] where \(x_0\) is an initial spatial shift, \(a\) is the initial width of the wave packet, and \(a\neq\sqrt{\hbar/m\omega}\) (which is the natural width of the oscillator ground state).
Tasks
Determine the time evolution of the wave packet, \(\psi(x,t)\).
Calculate the expectation values of position \(\langle x\rangle(t)\) and momentum \(\langle p\rangle(t)\).
Determine the uncertainty in position \(\Delta x(t)\) and show how the wave packet width oscillates or “breathes” over time.
Solution & Dynamics
Analytical Solution
1. Time Evolution of Expectation Values
Using Ehrenfest’s Theorem for a particle in a harmonic oscillator potential \(V(x) = \frac{1}{2} m \omega^2 x^2\), the expectation values obey classical equations of motion: \[ \frac{d\langle x \rangle}{dt} = \frac{\langle p \rangle}{m}, \quad \frac{d\langle p \rangle}{dt} = -m \omega^2 \langle x \rangle \] Solving these coupled differential equations with initial conditions \(\langle x \rangle(0) = x_0\) and \(\langle p \rangle(0) = 0\) yields: \[ \boxed{ \langle x \rangle(t) = x_0 \cos(\omega t) } \]\[ \boxed{ \langle p \rangle(t) = -m \omega x_0 \sin(\omega t) } \] The center of the wave packet oscillates harmonically with frequency \(\omega\).
2. Position Uncertainty and “Breathing”
For a Gaussian wave packet, the uncertainty in position \(\Delta x(t)\) depends on its initial width \(a\). If the initial width differs from the natural width of the oscillator ground state \(l = \sqrt{\hbar / m \omega}\), the state is “squeezed”. The time evolution of the variance is given by: \[ (\Delta x(t))^2 = \frac{a^2}{2} \cos^2(\omega t) + \frac{\hbar^2}{2 m^2 \omega^2 a^2} \sin^2(\omega t) \] Taking the square root gives the position uncertainty: \[ \boxed{ \Delta x(t) = \frac{a}{\sqrt{2}} \sqrt{ \cos^2(\omega t) + \left( \frac{l^2}{a^2} \right)^2 \sin^2(\omega t) } } \] This shows that the width of the wave packet oscillates (or “breathes”) with a frequency \(2\omega\). If \(a = l\), the width remains constant \(\Delta x(t) = l/\sqrt{2}\) (coherent state).
Computational Solution
We will compute the evolution numerically by expanding the Gaussian in the basis of harmonic oscillator eigenstates.
Left: Potential well V(x) along with the initial and half-period wavefunctions. Right: Analytical vs Computational expectation value and position uncertainty.
A particle of mass \(m\) moves in a one-dimensional quartic double-well potential described by: \[V(x)=-\frac{1}{2}\alpha x^2+\beta x^4\] where \(\alpha\) and \(\beta\) are positive constants.
Tasks
Find the exact energy eigenvalues for the ground state (\(E_0\)) and the first excited state (\(E_1\)) using computational methods.
Calculate the energy splitting \(\Delta E=E_1-E_0\).
Determine the tunneling time \(\tau\propto\frac{\hbar}{\Delta E}\) for a particle initially localized in the right well to tunnel to the left well.
Solution & Dynamics
Analytical Solution
For a quartic double-well potential \(V(x)=-\frac{1}{2}\alpha x^2+\beta x^4\), there are two minima at \(x_{min} = \pm \sqrt{\alpha/4\beta}\).
1. Degeneracy and Splitting
When the barrier between the wells is high, the lowest two energy levels are nearly degenerate. The true ground state \(\psi_0(x)\) is symmetric, while the first excited state \(\psi_1(x)\) is antisymmetric: \[ \psi_0(x) \approx \frac{1}{\sqrt{2}} (\phi_L(x) + \phi_R(x)) \]\[ \psi_1(x) \approx \frac{1}{\sqrt{2}} (\phi_L(x) - \phi_R(x)) \] where \(\phi_L\) and \(\phi_R\) are states localized in the left and right wells, respectively. The energy splitting \(\Delta E = E_1 - E_0\) is small and determines the tunneling dynamics.
2. Tunneling Time
If a particle is initially localized entirely in the right well, the initial state is a superposition: \[ \Psi(x,0) = \phi_R(x) = \frac{1}{\sqrt{2}} (\psi_0(x) - \psi_1(x)) \] The time evolution introduces a relative phase between the eigenstates: \[ \Psi(x,t) = \frac{1}{\sqrt{2}} \left( \psi_0(x) e^{-i E_0 t / \hbar} - \psi_1(x) e^{-i E_1 t / \hbar} \right) \] The particle completely tunnels to the left well when the relative phase reaches \(\pi\). This occurs at the tunneling time \(\tau\): \[ \frac{(E_1 - E_0)\tau}{\hbar} = \pi \implies \boxed{ \tau = \frac{\pi \hbar}{\Delta E} } \]
Computational Solution
To find the exact eigenvalues and eigenfunctions, we use the Finite Difference Method (FDM) to discretize the Schrödinger equation and turn it into a matrix eigenvalue problem (Hamiltonian diagonalization).
Left: Double well potential and the lowest two eigenstates (symmetric and antisymmetric). Right: Zoomed-in energy spectrum showing the splitting \(\Delta E\).
