Schrodinger Equation Wave Function Probability Density Potential Well Harmonic Oscillator Tunneling

# Schrödinger Dynamics, Bound States, Oscillators, and Tunneling

## 1. State vectors and wavefunctions

Nonrelativistic quantum mechanics represents a pure state by a vector in Hilbert space. In the position basis, its components form a complex wavefunction ψ(x,t). Global phase has no observable effect, but relative phase controls interference. The wavefunction is not a classical material wave; it is a probability amplitude from which measurement statistics follow.

The Born rule assigns probability density:

$$ P(x,t)=|\psi(x,t)|^2=\psi^*(x,t)\psi(x,t) $$

The probability of finding the particle in interval [a,b] is the integral of this density. A state representing one particle must be normalized:

$$ \int_{-\infty}^{\infty}|\psi(x,t)|^2dx=1 $$

Square-integrability, continuity, boundary conditions, and finite physical expectation values restrict acceptable states.

## 2. Operators and observables

An observable is represented by a Hermitian operator. In one dimension the position and momentum operators are:

$$ \hat x=x,\qquad \hat p=-i\hbar\frac{\partial}{\partial x} $$

They obey the canonical commutator:

$$ [\hat x,\hat p]=i\hbar $$

The expectation of observable A is:

$$ \langle A\rangle=\int\psi^*\hat A\psi\,dx $$

Hermiticity ensures real eigenvalues and orthogonality of distinct eigenstates. Measurement outcomes are operator eigenvalues with probabilities determined by projections of the state.

## 3. Building the Schrödinger equation

For a free classical particle, energy is p²/(2m). A plane matter wave has phase kx − ωt and de Broglie-Einstein relations p = ħk and E = ħω. Acting on a plane wave, iħ times the time derivative returns E, while −iħ times the spatial derivative returns p. Promoting the classical energy relation to operators gives the Hamiltonian:

$$ \hat H=-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V(x,t) $$

Time evolution is postulated to satisfy:

$$ i\hbar\frac{\partial\psi}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+V(x,t)\psi $$

This time-dependent Schrödinger equation is first order in time, so an initial normalized state determines later evolution. A Hermitian Hamiltonian produces unitary evolution and preserves normalization.

## 4. Probability conservation

Combine the Schrödinger equation with its complex conjugate to obtain a continuity equation:

$$ \frac{\partial |\psi|^2}{\partial t}+\frac{\partial j}{\partial x}=0 $$

The probability current is:

$$ j=\frac{\hbar}{2mi}\left(\psi^*\frac{\partial\psi}{\partial x}-\psi\frac{\partial\psi^*}{\partial x}\right) $$

Probability lost from a region equals current through its boundaries. This local conservation law also makes transmission and reflection calculations precise.

## 5. Stationary states

When V is time independent, separate variables with ψ(x,t) = φ(x)T(t). Division produces a constant E. The temporal factor is a phase exp(−iEt/ħ), while the spatial factor satisfies:

$$ -\frac{\hbar^2}{2m}\frac{d^2\phi}{dx^2}+V(x)\phi=E\phi $$

This is the time-independent Schrödinger equation. A stationary energy eigenstate has time-independent probability density because its time factor has unit magnitude. General states are superpositions:

$$ \psi(x,t)=\sum_n c_n\phi_n(x)e^{-iE_nt/\hbar} $$

Relative phases evolve and can make the density time dependent even though each component is stationary.

## 6. Infinite square well

Consider V = 0 for 0 < x < L and infinite elsewhere. The wavefunction vanishes at both walls. Inside, the equation has sinusoidal solutions. Boundary conditions select integer wave numbers:

$$ k_n=\frac{n\pi}{L},\qquad n=1,2,3,\ldots $$

Normalized eigenfunctions are:

$$ \phi_n(x)=\sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right) $$

Energies are quantized:

$$ E_n=\frac{n^2\pi^2\hbar^2}{2mL^2} $$

There is no n = 0 state because it would be the zero function and could not be normalized. Ground-state energy is nonzero, reflecting confinement and the uncertainty principle. The nth eigenfunction has n − 1 interior nodes.

## 7. Measurements and superpositions in the well

If ψ is a sum of energy eigenstates, an energy measurement returns En with probability |cn|². Position probability may oscillate through interference terms. Orthogonality gives coefficients:

$$ c_n=\int_0^L\phi_n^*(x)\psi(x,0)dx $$

Completeness means an acceptable initial state can be expanded in the eigenbasis. Parseval's relation makes the coefficient probabilities add to one.

## 8. Finite wells

A finite well allows exponential tails into classically forbidden regions. Bound-state energies remain discrete but are lower in number than for an infinite well. Wavefunction and its derivative are continuous at finite potential steps. Even and odd parity simplify symmetric wells. Energy conditions are transcendental and generally solved numerically.

As well depth or width decreases, excited bound states disappear into the continuum. In one dimension an attractive finite well supports at least one bound state under broad conditions.

## 9. Quantum harmonic oscillator

For a quadratic potential:

$$ V(x)=\frac{1}{2}m\omega^2x^2 $$

Introduce dimensionless combinations of position and momentum through lowering and raising operators:

$$ \hat a=\sqrt{\frac{m\omega}{2\hbar}}\hat x+\frac{i}{\sqrt{2m\hbar\omega}}\hat p $$
$$ \hat a^\dagger=\sqrt{\frac{m\omega}{2\hbar}}\hat x-\frac{i}{\sqrt{2m\hbar\omega}}\hat p $$

Canonical commutation implies:

$$ [\hat a,\hat a^\dagger]=1 $$

The Hamiltonian becomes:

$$ \hat H=\hbar\omega\left(\hat a^\dagger\hat a+\frac{1}{2}\right) $$

The number operator has nonnegative integer eigenvalues. Ladder action changes energy by one quantum:

$$ \hat a|n\rangle=\sqrt n|n-1\rangle,\qquad \hat a^\dagger|n\rangle=\sqrt{n+1}|n+1\rangle $$

Therefore:

$$ E_n=\hbar\omega\left(n+\frac{1}{2}\right) $$

## 10. Zero-point motion

The oscillator ground state has energy ħω/2 and cannot sit motionless at x = p = 0, which would violate uncertainty. Requiring a|0⟩ = 0 yields a Gaussian:

$$ \phi_0(x)=\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}e^{-m\omega x^2/(2\hbar)} $$

Excited states are this Gaussian multiplied by Hermite polynomials. Parity alternates, and the nth state has n nodes. Position and momentum uncertainties in the ground state saturate the uncertainty bound.

## 11. Classical and quantum oscillator limits

At large n, probability accumulates near classical turning points because classical motion is slow there. Quantum density still oscillates rapidly, but coarse-grained averages approach the classical distribution. Coherent states are minimum-uncertainty wave packets whose centres follow classical oscillator trajectories while retaining quantum spread.

## 12. Potential steps

For a step with E greater than V₀, both regions support travelling waves but their wave numbers differ. Matching ψ and its derivative generates reflection even though the particle has enough classical energy to cross. Probability-current ratios define R and T, and for a real conservative step R + T = 1.

For E below V₀, the wavefunction in the high-potential region decays with:

$$ \kappa=\sqrt{\frac{2m(V_0-E)}{\hbar^2}} $$

A semi-infinite forbidden region carries no transmitted steady current, but a finite barrier permits a second interface and nonzero transmission.

## 13. Rectangular-barrier tunneling

For a barrier of height V₀ and width L with E < V₀, solve oscillatory waves outside and growing-decaying exponentials inside, then match ψ and dψ/dx at both interfaces. The exact transmission is:

$$ T=\left[1+\frac{V_0^2\sinh^2(\kappa L)}{4E(V_0-E)}\right]^{-1} $$

For an opaque barrier κL much larger than one:

$$ T\approx16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)e^{-2\kappa L} $$

The exponential makes tunneling extremely sensitive to mass, barrier width, and barrier energy. Electrons tunnel readily on nanometre scales; macroscopic objects do not.

## 14. Applications of tunneling

Alpha decay, scanning tunneling microscopy, tunnel diodes, Josephson junctions, flash-memory programming, and fusion all involve barrier penetration. The scanning tunneling current changes exponentially with tip-sample distance, enabling atomic-scale height sensitivity. Tunneling does not mean a particle borrows energy in a classical accounting violation; total energy remains well defined in a stationary scattering state.

## 15. Wave packets and dispersion

A localized free particle requires a superposition of momentum eigenstates:

$$ \psi(x,t)=\frac{1}{\sqrt{2\pi}}\int A(k)e^{i(kx-\omega(k)t)}dk $$

For nonrelativistic dispersion ω = ħk²/(2m), different k components acquire phase at different rates, so a Gaussian packet spreads. Ehrenfest's theorem shows expectation values obey Newton-like equations:

$$ m\frac{d^2\langle x\rangle}{dt^2}=-\left\langle\frac{dV}{dx}\right\rangle $$

Classical motion emerges when the packet remains narrow on the scale over which the force changes.

## 16. Numerical finite differences

On a grid with spacing Δx, approximate the second derivative by three points. The Hamiltonian becomes a sparse tridiagonal Hermitian matrix. Its lowest eigenpairs approximate bound energies and wavefunctions. Convergence requires shrinking Δx, moving boundaries outward, and checking normalization and orthogonality. Discontinuous potentials converge more slowly near interfaces.

The discrete kinetic matrix has diagonal ħ²/(mΔx²) and nearest-neighbour entries −ħ²/(2mΔx²). Boundary choices encode physics and must be reported.

## 17. Python eigenstate and tunneling solver

This program solves finite-well and harmonic-oscillator eigenstates with a sparse Hamiltonian, compares oscillator energies with theory, and plots exact and opaque-barrier transmission. SI units are used internally and electron volts for presentation.

import numpy as np
import matplotlib.pyplot as plt
from scipy.constants import hbar, m_e, e
from scipy.sparse import diags
from scipy.sparse.linalg import eigsh

def solve_1d(x, potential_ev, mass=m_e, states=6):
    dx = x[1] - x[0]
    kinetic_scale = hbar**2 / (2 * mass * dx**2)
    main = 2 * kinetic_scale + potential_ev * e
    off = -kinetic_scale * np.ones(len(x) - 1)
    hamiltonian = diags([off, main, off], [-1, 0, 1], format="csc")
    values, vectors = eigsh(hamiltonian, k=states, which="SA")
    order = np.argsort(values)
    values, vectors = values[order], vectors[:, order]
    for index in range(states):
        norm = np.sqrt(np.trapezoid(abs(vectors[:, index])**2, x))
        vectors[:, index] /= norm
    return values / e, vectors

x = np.linspace(-3e-9, 3e-9, 1600)
finite_well = np.where(abs(x) < 1.2e-9, 0.0, 2.0)
well_energy, well_psi = solve_1d(x, finite_well, states=5)

omega = 2.2e15
oscillator = 0.5 * m_e * omega**2 * x**2 / e
osc_energy, osc_psi = solve_1d(x, oscillator, states=5)
osc_exact = hbar * omega / e * (np.arange(5) + 0.5)

def barrier_transmission(energy_ev, height_ev, width_m, mass=m_e):
    energy = np.asarray(energy_ev) * e
    height = height_ev * e
    below = energy < height
    result = np.empty_like(energy)
    kappa = np.sqrt(2 * mass * np.maximum(height - energy, 0)) / hbar
    denominator = 1 + height**2 * np.sinh(kappa[below] * width_m)**2 / (
        4 * energy[below] * (height - energy[below]))
    result[below] = 1 / denominator
    q = np.sqrt(2 * mass * np.maximum(energy[~below] - height, 0)) / hbar
    denominator_high = 1 + height**2 * np.sin(q * width_m)**2 / (
        4 * energy[~below] * (energy[~below] - height) + 1e-300)
    result[~below] = 1 / denominator_high
    return result

energy_grid = np.linspace(0.01, 2.8, 1000)
transmission = barrier_transmission(energy_grid, 2.0, 0.8e-9)

fig, axes = plt.subplots(1, 3, figsize=(16, 4.5))
axes[0].plot(x * 1e9, finite_well, "k", label="V")
for n in range(4):
    axes[0].plot(x * 1e9, well_energy[n] + 0.15 * well_psi[:, n] / np.max(abs(well_psi[:, n])),
                 label=f"n={n}, E={well_energy[n]:.3f} eV")
axes[0].set(xlabel="x (nm)", ylabel="energy (eV)", title="Finite-well eigenstates")
axes[0].legend(fontsize=7)

axes[1].plot(x * 1e9, oscillator, "k", label="V")
for n in range(4):
    axes[1].plot(x * 1e9, osc_energy[n] + 0.12 * osc_psi[:, n] / np.max(abs(osc_psi[:, n])))
axes[1].set(xlabel="x (nm)", ylabel="energy (eV)", title="Harmonic oscillator")
axes[1].set_ylim(0, osc_energy[4] * 1.4)

axes[2].semilogy(energy_grid, transmission)
axes[2].axvline(2.0, color="gray", linestyle="--", label="barrier height")
axes[2].set(xlabel="particle energy (eV)", ylabel="transmission",
            title="Rectangular-barrier transmission", ylim=(1e-8, 1.2))
axes[2].legend()
axes[2].grid(True, which="both")
plt.tight_layout()
plt.show()

print("Finite-well energies (eV):", well_energy)
print("Numerical oscillator energies:", osc_energy)
print("Exact oscillator energies:", osc_exact)
print("Relative oscillator errors:", (osc_energy - osc_exact) / osc_exact)

## 18. Validation checklist

Verify Hamiltonian Hermiticity, normalized eigenvectors, orthogonality, stable low energies under grid refinement, and negligible boundary amplitude for bound states. Count nodes and compare parity. For scattering, confirm 0 ≤ T ≤ 1 and R + T = 1. Test that transmission decreases with width and approaches unity in appropriate high-energy limits.

## 19. Common errors

  • Probability density is |ψ|², not ψ itself.
  • A stationary state can have nonzero probability current.
  • Infinite-well quantum number starts at one.
  • Energy quantization follows boundary conditions, not an arbitrary rounding rule.
  • Barrier penetration is not classical energy violation.
  • A finite-difference eigenvalue is meaningful only after convergence checks.
  • Degenerate subspaces may rotate numerically while representing the same physics.

## 20. Final perspective

The Schrödinger equation makes quantum dynamics deterministic at the level of amplitudes while measurements remain probabilistic. Boundary conditions create discrete bound spectra; algebra exposes the oscillator ladder and zero-point energy; matching conditions create tunneling. Analytical structure and numerical computation reinforce each other when normalization, current, units, and convergence are checked explicitly.

## 21. Variational and uncertainty estimates

Exact solutions are rare, but bounds and scaling remain powerful. For any normalized trial state, the Hamiltonian expectation cannot lie below the true ground energy:

$$ E_0\leq\frac{\langle\psi_t|\hat H|\psi_t\rangle}{\langle\psi_t|\psi_t\rangle} $$

Choose a trial family with adjustable width, calculate kinetic and potential expectations, and minimize. Narrow localization raises kinetic energy roughly as ħ²/(mℓ²), while confinement potential often rises for broad states. Their competition estimates ground-state size and energy before numerical diagonalization.

This scaling also explains mass dependence: heavier particles pay less kinetic energy for localization and tunnel less through the same barrier. Comparing variational estimates, grid eigenvalues, and known limits is a strong three-way validation.

## 22. Time propagation

Eigenvalue solvers describe stationary structure. Dynamics can be computed with a unitary method such as Crank-Nicolson or split-operator Fourier propagation. A trustworthy propagation monitors norm, mean energy for a time-independent Hamiltonian, boundary reflections, and time-step convergence. Absorbing layers mimic open boundaries but intentionally remove norm, so lost probability must equal outgoing flux.

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