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Topic 3 of 12

Schrödinger Equations

The Schrödinger equation is really a family of eigenvalue equations — the same underlying idea written in whichever basis is most convenient.

One example of the Schrödinger equation is

H^∣ψ⟩=iℏ∂∂t∣ψ⟩\hat{H} \ket{\psi} = i \hbar \frac{\partial }{\partial t}\ket{\psi}

However, another example is

p^∣ψ⟩=−iℏ∂∂x∣ψ⟩\hat{p} \ket{\psi} = - i \hbar \frac{\partial}{\partial x}\ket{\psi}

Bases

There are many different Schrödinger equations. They are eigenvalue equations. The basis is on what ψ\psi depends.

Independent basis

p^∣ψ⟩=a∣ψ⟩\hat{p} \ket{\psi} = a\ket{\psi}
x^∣ψ⟩=b∣ψ⟩\hat{x} \ket{\psi} = b\ket{\psi}
H^∣ψ⟩=c∣ψ⟩\hat{H} \ket{\psi} = c\ket{\psi}

Time basis

H^∣ψ⟩=iℏ∂∂t∣ψ⟩\hat{H} \ket{\psi} = i \hbar \frac{\partial }{\partial t}\ket{\psi}

Position basis

x^∣ψ⟩=x∣ψ⟩\hat{x} \ket{\psi} = x \ket{\psi}
p^∣ψ⟩=−iℏ∂ψ∂x\hat{p} \ket{\psi} = - i \hbar \frac{\partial \psi}{\partial x}
T^∣ψ⟩=−ℏ22m∂2ψ∂x2\hat{T} \ket{\psi} = \frac{-\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2}
V^∣ψ⟩=V(x)∣ψ⟩\hat{V} \ket{\psi} = V(x) \ket{\psi}
H^∣ψ⟩=−ℏ22m∂ψ∂x2+V(x)ψ(x)\hat{H} \ket{\psi} = -\frac{\hbar^2}{2m} \frac{\partial \psi}{\partial x^2} + V(x) \psi(x)

Momentum basis

p^∣ψ⟩=p∣ψ⟩\hat{p} \ket{\psi} = p \ket{\psi}
x^∣ψ⟩=iℏ∂ψ∂p\hat{x} \ket{\psi} = i \hbar \frac{\partial \psi}{\partial p}

Spin basis

Content coming soon.

Energy basis

H^∣ψ⟩=E∣ψ⟩\hat{H} \ket{\psi} = E \ket{\psi}
x^∣n⟩=ℏ2mω(a+a†)∣n⟩\hat{x} \ket{n} = \sqrt{\frac{\hbar}{2m \omega}} \biggr(a + a^\dagger \biggr) \ket{n}

Time independent Schrödinger Equation

H^ψ=(T^+V^)ψ=T^ψ+V^ψ\hat{H} \psi = (\hat{T} + \hat{V}) \psi = \hat{T} \psi + \hat{V} \psi

Stationary States

Content coming soon.

Independent Basis

Content coming soon.

Position basis

−ℏ22m∇2ψ(r)+Vψ(r)-\frac{\hbar^2}{2m} \nabla^2 \psi(\boldsymbol{r}) + V \psi(\boldsymbol{r})

Momentum basis

Content coming soon.

Spin basis

Content coming soon.

WKB Approximation

Content coming soon.

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)

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4 of 7

For ψ(x) = A e^{ikx} with k=5.0 nm⁻¹, apply the momentum operator p̂=-iℏd/dx. What is the momentum eigenvalue in kg·m/s? (ℏ=1.055×10⁻³⁴)

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5 of 7

A photon has wavelength 500 nm. Find its energy in eV.

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6 of 7

An electron is in a superposition ψ = (1/√2)(ψ₁ + ψ₂) where E₁=1 eV, E₂=3 eV. Find ⟨E⟩ in eV.

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7 of 7

A particle of mass 1.0×10⁻²⁷ kg is confined to a region of size 1.0 nm. Estimate its minimum kinetic energy in eV using the uncertainty principle.

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Key Takeaways

  • The Schrödinger equation takes a different explicit form in every basis — position, momentum, energy, time
  • The time-independent equation H^ψ=Eψ\hat H \psi = E\psi is an eigenvalue equation for the Hamiltonian
  • The wave vector k relates to momentum via p = ℏk
  • Stationary states time-evolve only in phase
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