Schrödinger Equation

The Schrödinger equation governs the evolution of quantum states. The time-independent version yields energy eigenstates; the time-dependent version governs all dynamics.

Key Concepts

Time-dependent: iℏ ∂ψ/∂t = Ĥψ
Time-independent: Ĥψ = Eψ (eigenvalue equation)
Hamiltonian operator: Ĥ = -ℏ²/2m ∇² + V(x)
Energy eigenstates evolve as e^{-iEt/ℏ}
Superposition of energy eigenstates gives general time evolution

Key Equations

Time-dependent SE

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

Time-independent SE

-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V\psi = E\psi

Stationary state

\Psi(x,t) = \psi(x)e^{-iEt/\hbar}

de Broglie relation

p = \hbar k,\quad E = \hbar\omega

Worked Example

Example Problem

Problem

An electron (m=9.11×10⁻³¹ kg) has kinetic energy 1.0 eV in a region of zero potential. Find its wave vector k.

Solution

E = ℏ²k²/2m → k = √(2mE)/ℏ. E=1.0 eV = 1.6×10⁻¹⁹ J. k = √(2×9.11×10⁻³¹×1.6×10⁻¹⁹)/(1.055×10⁻³⁴) = √(2.92×10⁻⁴⁹)/1.055×10⁻³⁴ ≈ 5.12×10⁹ m⁻¹.

Practice

Exercises

7 problems

1 of 7

An electron with kinetic energy 4.0 eV moves in free space. Observe the animated de Broglie wave and use the ruler to determine the wave vector k in nm⁻¹.

The animation shows ψ(x,t) = A·exp(ikx − iωt) for a 4 eV electron. The yellow bracket marks one wavelength λ. Use λ to find k = 2π/λ in nm⁻¹.

k = nm⁻¹

2 of 7

For the same 4.0 eV electron, use the wavelength ruler on the wave visualization to find its de Broglie wavelength λ in nm.

Drag the green (A) and amber (B) markers to two adjacent crests of the blue wave. Read off λ, then submit.

Measured λ = 0.614 nm

λ = nm

3 of 7

A particle in a stationary state has energy E = 2.0 eV. Find the angular frequency ω in rad/s. (ℏ=1.055×10⁻³⁴ J·s)

Unlock Exercise 3