Quantum Harmonic Oscillator

Visualizing using Numerical Methods

WIP

Figures

Figure 1: 2D Plot of ∣Ψ∣|\Psi| vs xx, Animated by Time

.

This figure shows the absolute value of the wave function as a function of position. It is animated over time to illustrate the evolution of the state.

Figure 2: 3D Plot of Re(Ψ)\text{Re}(\Psi), Im(Ψ)\text{Im}(\Psi), xx, Animated by Time

. This figure provides a 3D visualization of the real and imaginary parts of the wave function. The animation shows how these components change over time.

Figure 3: 3D Plot of Eigenstates and Their Rotation as a Function of Time

. This figure depicts the eigenstates of the Hamiltonian. It shows how these states rotate over time while maintaining their orthonormality.

  • Time Dependent Shrodinger Equation
    • iℏ∂∣Ψ⟩∂t=H^∣Ψ⟩i \hbar \frac{\partial \ket{\Psi } }{\partial t} = \hat{H} \ket{\Psi }
    • H^\hat{H} is the hamiltonian
      • H^=−ℏ22m∇2+V^\hat{H} = -\frac{\hbar^2}{2m} \nabla ^2 + \hat{V}
  • Basis
    • we use the descrete orthonormal basis. the basis is formed by the eigenstates of the position operator X^\hat{X}
    • Construction
      • split the interval [0,1]\left[ 0,1 \right] into NN points
        • Δx=1N−1\Delta x = \frac{1}{N-1}
        • xi=i⋅Δx∀i∈{0,…,1}x_{i}=i\cdot \Delta x \quad \forall i \in \left\{ 0,\ldots ,1 \right\}
      • dot product
        • ⟨i∣j⟩=δij\braket{i|j}=\delta _{ij}
    • Representation
      • ∣i⟩\ket{i} represents the point at xix_{i}
      • can be represented as the column vector (0…1…0)\left( 0 \ldots 1 \ldots 0 \right), where the 11 is at the ith position
    • any state Ψ(x)\Psi(x) can be represented as ∑n=0N−1cn∣n⟩\sum_{n=0}^{N-1} c_n \ket{n} where cn=Ψ(xn)c_n = \Psi(x_n)
    • any operator O^\hat{O} can be represented in the form of a matrix
      • Oij=⟨i∣O^∣j⟩O_{ij} = \bra{i} \hat{O} \ket{j}
      • Eg: Identity I^\hat{I}
        • Iij=⟨i∣I^∣j⟩=⟨i∣j⟩=δijI_{ij} = \bra{i} \hat{I} \ket{j} = \braket{i|j} = \delta _{ij}
        • this is the diagonal matrix with all diagonal entries 1
        • this also proves the completeness of our basis
      • Eg: Position X^\hat{X}
        • Xij=⟨i∣X^∣j⟩X_{ij}=\bra{i}\hat{X}\ket{j}
        • as X^∣j⟩=xj∣j⟩\hat{X}\ket{j} = x_j \ket{j}
        • Xij=xj⟨i∣j⟩=xjδijX_{ij}=x_j \braket{i|j}= x_j \delta _{ij}
        • this is a diagonal matrix with the positions as diagonal entries
  • Harmonic Potential
    • what is a harmonic oscillator
    • why is studying it useful
    • V(x)=12mω2(x−12)2V(x)=\frac{1}{2}m \omega ^2 (x-\frac{1}{2})^2
    • V^=12mω2[X^−I^/2]2\hat{V}=\frac{1}{2}m \omega ^2 \left[\hat{X}-\hat{I}/2\right]^2
  • Laplacian operator ∇2^\hat{\nabla ^2}
    • To approximate the laplacian in our basis, we can use the secant-line method
    • ∂2∂x2≈xi+1+xi−1−2xiΔx2\frac{\partial ^2 }{\partial x^2} \approx \frac{x_{i+1}+x_{i-1}-2x_{i}}{{\Delta x} ^2}
    • ∇2^≈∑n∣n+1⟩⟨n∣+∣n−1⟩⟨n∣−2∣n⟩⟨n∣Δx2\hat{\nabla ^2} \approx \sum_n \frac{\ket{n+1}\bra{n}+\ket{n-1}\bra{n}-2\ket{n}\bra{n}}{{\Delta x}^2}
  • We can combine the potential and kinetic energy terms to get the hamiltonian H^\hat{H}
  • We can then use Euler’s method to evolve the state function
    • ∣Ψ′⟩=(I^−iℏΔtH^)∣Ψ⟩\ket{\Psi'}=\left( \hat{I} - \frac{i}{\hbar}\Delta t \hat{H} \right)\ket{\Psi }
  • But we can do better
    • as our hamiltonian is time independent, we can evaluate ∣Ψ(t)⟩\ket{\Psi (t)} directly given ∣Ψ(0)⟩\ket{\Psi (0)}
    • From TDSE ∂∣Ψ⟩∂t=−iℏH^∣Ψ⟩\frac{\partial \ket{\Psi }}{\partial t} = - \frac{i}{\hbar} \hat{H} \ket{\Psi}
    • d∣Ψ⟩∣Ψ⟩=−idtℏH^\frac{\mathrm{d }\ket{\Psi }}{\ket{\Psi }} = -\frac{i \mathrm{dt}}{\hbar} \hat{H}