Quantum Methods with Mathematica

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Assume a wavefunction of the form **psi[x, t] == f[t] psi[x]** and perform a separation of variables on the wave equation.

Show that **f[t] = E^(-I w t)** where **h w** is the separation constant. Try the built-in function **DSolve**.

Equate **h w** to the **Energy** by evaluating the [expected] value of **hamiltonian[V]** in the state **psi[x, t]**.

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Remove["Global`*"] hbar:=\[HBar] H[V_] @psi_:=-hbar^2/(2m) D[psi,{x,2}] + V psi psi[x_,t_]:=f[t] psi[x] I hbar D [psi[x,t],t]==H[V] @ psi[x, t] I hbar D [psi[x,t],t] / psi[x,t]==H[V] @ psi[x,t] / psi[x,t]

E1:=I hbar D [psi[x,t],t] / psi[x,t]==H[V] @ psi[x,t] / psi[x,t] Simplify[E1]

E2:=- 1/2 hbar hbar (D[D[psi[x],x],x]/(m psi[x]))==hbar omega DSolve[E2, psi[x], x] E3:=1/2 hbar 2 i D[f[t],t] / f[t]==hbar omega DSolve[E3, f[t], t]

k psi[x_]:=c E^(I k x) psi[x] f[t_]:=E^(-I omega t) f[t] psi[x_,t_]:=f[t] psi[x] psi[x,t]

E4:=Conjugate[psi[x,t]] H[0] @ psi[x,t] E4 E5:=Simplify[E4] E5 k:=Sqrt[2 m omega / hbar] Refine[E5, {Element[{c, omega, m, t, hbar, k, x}, Reals]}]

E6:=Conjugate[psi[x,t]] psi[x,t] Simplify[E6]

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— Me@2022-11-26 07:17:29 PM

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2022.11.28 Monday (c) All rights reserved by ACHK