Variational method

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Record your ground state
energy estimates

Gaussian:

Super-Gaussian 1:

Super-Gaussian 2:

Displaced Gaussian:

⟨H⟩_min = ... ħω

Energy vs. α graph for V = kx²/2

α

The variational method allows you to find an approximation to the ground state eigenfunction and energy using a trial function that depends on a parameter α. Note the bottom middle graph shows you ψ as a function of x, but the top graph energy expectation values as a function of α (not x). Find the ground state energy estimates for the different trial functions, and use the drop-down panels to record them and to change potential energies. Press ? for information. Then go on to the Challenges!

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Potential energy

V = kx²/2

x

Trial wave function ψ_α(x)

x

Change parameter α

small α

large α

Choose trial wave function

Trial function ψ for ground state energy estimate

Gaussian

Super-Gaussian 1

Super-Gaussian 2

Displaced Gaussian

ψ₁ = A_α e ^{– (x /α) ²} ψ₂ = A_α e ^{– (x /α) ⁴} ψ₃ = A_α e ^{– (x /α) ⁶} ψ₄ = A_α e ^{– ((x-μ) /α) ²}