Quantum Postulates and One-Dimensional Systems

Quantum wavefunctions and energy spectra

A stationary state of a bound particle is an eigenfunction of the Hamiltonian, H\psi_n = E_n\psi_n, and its energy spectrum depends on the shape of the potential and the boundary conditions. For a particle in a one-dimensional infinite well of width a, the wavefunction must vanish at the walls, which gives standing waves \psi_n \propto \sin(n\pi x/a) and levels E_n = n^2\pi^2\hbar^2/(2ma^2), so the spacing between levels grows with n^2. When a state is written as a superposition \psi = \sum_n c_n\psi_n, the probability of finding the energy E_n is |c_n|^2, the coefficients satisfy \sum_n |c_n|^2 = 1 by normalization, and the expectation value is \langle E\rangle = \sum_n |c_n|^2 E_n. A plane wave \Psi = Ae^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)} propagates along the wave vector \mathbf{k}, so its direction of propagation is the unit vector \hat{k} = \mathbf{k}/|\mathbf{k}|, which is found by reading the components of \mathbf{k} from the exponent and dividing by |\mathbf{k}|. For several identical non-interacting particles, the total energy is the sum of the single-particle energies of the occupied states, so excited configurations are found by listing the allowed sums of squares of the quantum numbers in increasing order. A superposition of two stationary states of energies E_a and E_b acquires a relative phase (E_b-E_a)t/\hbar, so the overlap between the state at t=0 and the state at time t can vanish when this phase equals an odd multiple of \pi. The wavefunction is continuous, and its first derivative is continuous wherever the potential is finite; it must vanish where the potential is infinite. In a classically forbidden region with E<V_0 the solution is evanescent, obeying \psi'' = \frac{2m(V_0-E)}{\hbar^2}\psi, so where \psi>0 it is convex and decays without an interior maximum. When a small extra potential V_0 \ll E is present in a region, first-order perturbation theory gives an energy shift \Delta E \approx V_0 P, where P is the probability of finding the particle in that region in the unperturbed ground state. For a two-dimensional square infinite well the energies are E = (n_x^2+n_y^2)E_0, and the interior nodal lines of \sin(n_y\pi y/L) lie at y = kL/n_y for integer k with 0<k<n_y, so counting the nodal lines fixes the quantum numbers. The radial equation for an s-state in a spherical region can be simplified by writing R = u(r)/r, which turns the radial Laplacian \frac{1}{r^2}\frac{d}{dr}\!\left(r^2\frac{dR}{dr}\right) into a one-dimensional second derivative in u with u(0)=0, so the ground state is u \propto \sin(\pi r/a) for a shell of radius a. When a hard wall at x=0 sits on a harmonic potential, only the odd-parity states of the full oscillator survive because their wavefunctions vanish at the wall, so the levels are \hbar\omega(n+\tfrac12) with odd n only. A constant offset V_0 shifts every level equally, while the oscillator frequency sets the spacing, so energies of such half-oscillator configurations take the form V_0+\hbar\omega_0(2j+\tfrac32) for j=0,1,2,\ldots, and both the offset and the frequency must be included when comparing configurations. For a three-dimensional isotropic oscillator, the energy is E_n = (n+\tfrac32)\hbar\omega with n = n_x+n_y+n_z, and the degeneracy of level n is the number of ways to distribute n among three quantum numbers, d_n = (n+1)(n+2)/2.

Harmonic oscillator ladder operators

The ladder operators a and a^\dagger factorize the oscillator Hamiltonian as H = \hbar\omega(a^\dagger a + \tfrac12). The creation operator raises the level, a^\dagger|n\rangle = \sqrt{n+1}\,|n+1\rangle, and the annihilation operator lowers it, a|n\rangle = \sqrt{n}\,|n-1\rangle. Position and momentum are linear combinations of these operators, \hat{x} = \sqrt{\hbar/2m\omega}\,(a+a^\dagger) and \hat{p} = i\sqrt{m\hbar\omega/2}\,(a^\dagger-a), so a single power of \hat x connects only neighbouring levels and its matrix in the energy basis is tridiagonal with zero diagonal, with off-diagonal entries \sqrt{n+1} between |n\rangle and |n+1\rangle in the common prefactor. A product of two linear operators changes the occupation number by the sum of the individual shifts. Terms containing a^{\dagger 2} or a^2 change n by \pm 2, while the combination a^\dagger a - a a^\dagger = -1 is diagonal. Hence \hat p\hat x = \frac{i\hbar}{2}\left(a^{\dagger 2} - a^2 - 1\right) has nonzero matrix elements only between |n\rangle and |n\rangle or |n\pm 2\rangle, and never between levels that differ by one.

Energy-time uncertainty for photons

The energy-time uncertainty relation \Delta E\,\Delta t \gtrsim \hbar/2 connects the duration of a pulse to the spread of its energy. For a photon the energy is E = pc, so the momentum spread is \Delta p = \Delta E/c. A pulse of duration \Delta t occupies a spatial length c\,\Delta t along its direction of propagation, so in order of magnitude the momentum uncertainty is \Delta p \approx h/(c\,\Delta t), and the units check out because h is in J s, c\,\Delta t is in m, and momentum is in kg m s^{-1}. Such questions are normally answered by estimating the order of magnitude with powers of ten, so the result is reported as an exponent.

Electron double-slit interference

In a double-slit experiment with particles of momentum p, the de Broglie wavelength is \lambda = h/p, but fringes are observable only if the beams leaving the two slits overlap on the screen. A slit of width w spreads the beam by an angle of order \lambda/w, so the spreading must be large enough compared with the slit separation d for the two beams to overlap at the screen distance D. In the form tested, the fringe condition is \hbar d w > pD, that is p < \hbar d w/D, meaning the momentum must be small enough, and the wavelength large enough relative to the slit geometry, for interference to be seen.