Quantum Mechanics

An introduction to, and derivation of, the fundamental equations of Schrodinger’s quantum mechanics

Tyson Jones, WSS17
tyson.jones@monash.edu

To smoothly evaluate this notebook, please (through the Evaluation menu):• Disable Dynamic Updating • Evaluate Notebook (yes to automatically evaluate all initialization cells)• Enable Dynamic UpdatingPre-evaluation is required to view the random-sampling plots

Note: we’ll work in natural and non-dimensional units where

ℏ=m=ω=1

Initialisation

ClearAll["Global`*"]

AutoCollapse[] := ( If[$FrontEnd =!= $Failed, SelectionMove[EvaluationNotebook[], All, GeneratedCell]; FrontEndTokenExecute["SelectionCloseUnselectedCells"]])

colorBar[title_:"arg[ψ]"] := BarLegend[ {"Rainbow", {-π, π}}, LegendLabel  title, "Ticks"  {-3.14, -3.14/2, 0, 3.14/2, 3.14}, "TickLabels"  {"-π", "-π/2", "0", "π/2", "π"} ]colorBarNorm[title_] := BarLegend[ "Rainbow", LegendLabel  title ]plotWavefunction[psi_, {r_, range__}, showbar_:True, title_:

Abs[ψ]

, legendpsi_:ψ, plotrange_:All, ticks_:Automatic] := ReplaceAll[ Plot[

Abs[psi]

, {r, range}, AxesLabel  {r, title}, PlotRange  plotrange, ColorFunction  (ColorData["Rainbow"][Rescale[Arg[psi /. r  #], {-π, π}]] &), ColorFunctionScaling  False, Ticks  ticks, Filling  Axis, PlotLegends  If[showbar, colorBarNorm["arg[" <> ToString[legendpsi] <> "] / 2π"], None] ], Line[pts_, _]  {Black, Line[pts]} ]

simulateWavefunction[psi_, potential_, {r_, domain__}, {_, times__}] := NDSolveValue  

(0,1)

[r, τ]  -

(2,0)

[r, τ]  2 + ψ[r, τ] potential, ψ[r, 0]  psi , ψ, {r, domain}, {τ, times}, Method  {"FiniteElement"}  Off[NDSolveValue::bcart]Off[NDSolveValue::femcscd]

hamiltonian[psi_, potential_, r_] := -

D[psi, {r,2}] + potential psi

expectedEnergy[psi_, potential_, r_] := Integrate[ (Conjugate[psi] /. Conjugate[r]  r) (hamiltonian[psi, potential, r]), {r, -∞, ∞} ]

quickNormalise[psi_, r_:x] := psi  Sqrt[NIntegrate[

Abs[psi]

, {r, -∞, ∞}, Method  {Automatic, "SymbolicProcessing"  0}]]

The Wavefunction

The quantum wavefunction is a (L2 normalizable) complex field defined over some continuous space of states. For example, consider a 1D wavefunction of position:

exampleψ[x_]=

(-1+x)

+2πx







(x-2)



1/4

//quickNormalise;

	Re(ψ)
	Im(ψ)

Although the wavefunction describes a physical system, it is itself unphysical. Only its absolute value squared can be observed.

	2 ψ
	Arg[ψ] / π

In the Born interpretation, the absolute value squared of the (normalised) wavefunction is a probability density function. If our wavefunction described a particle’s position, an integrated region of the squared norm would give the probability of finding the particle within that region when measured.

Pr(2 < x < 3) =

∫

|ψ|^2 x = 0.196048

Alternatively, if we measured the position of many particles all described by the same wavefunction, we’d expect the distribution to follow the norm squared of that wavefunction.

The probability density function alone however is an incomplete description of the system. The complex phase is very important; we’ll here-forth indicate it by colour.

The complex phase, together with the amplitude, govern the evolution of the wavefunction in time. A phase gradient indicates a probability current; that the probability density function (and the likelihood our particle will appear in a given region) will change in time.

This was an example of a free system, in no external potential. Its evolution, and the evolution of general trapped systems, is determined by the Schrodinger equation.

The Schrodinger Equation

Time dependent and independent forms

where r can represent one or more spatial dimensions.

We can often produce an infinite set of eigenfunctions ϕ[r,t] (with eigenvalues Ε) of the Hamiltonian, which each satisfy

and which can be expressed in terms of a time-independent eigenfunction ϕ[r] and an oscillator time component.

Returning this form to the Schrodinger equation gives

which for a linear and time-independent Hamiltonian, presents the time-independent Schrodinger equation of the energy eigenfunctions ϕ[r]

Operators and eigenfunctions

where the Laplacian gives an analog of kinetic energy. The time-dependent Schrodinger equation becomes

and the time-independent Schrodinger equation of the eigenfunctions becomes

Since the Schrodinger equation is linear, the total wavefunction evolves like the sum of the simple eigenfunction dynamics,

The quantum harmonic oscillator is an example of a simple Hamiltonian system with an interesting family of eigenfunctions, which can be used to propagate a wavefunction through time without numerically solving the time-dependent Schrodinger equation.

The Quantum Harmonic Oscillator

Finding the eigenfunctions

Consider a simple one-dimensional quadratic trapping potential.

The corresponding time-independent Schrodinger equation is

Since Hermite polynomials must have integer degree, we’ve found the discretized energy eigenvalues.

so our eigenfunctions become

The colours here correspond to 0 and -π phase; that the wavefunction is positive and negative real.

Superposing the eigenfunctions

Mixing out-of-phase eigenfunctions will introduce phase gradients...

which are otherwise induced in-time by mixing in-phase eigenfunctions, due to their different eigenvalues.

These examples (composed of a finite number of eigenfunctions) are periodic at some multiple of π. Even finite superpositions can lead to complicated (but periodic) evolution.

By forming a complete, orthonormal basis, the energy eigenfunctions can be linearly superposed to produce any possible wavefunction, though this generally requires non-zero coefficients of an infinite number of them (a rather hard computation).

Coherent States

A coherent wavefunction is one which maintains its probability density profile in time while still translating in space.

We created the above (due to numerical imprecision; approximate) coherent wavefunction of the quantum harmonic oscillator by translating the lowest energy eigenfunction (the “ground-state”) out of the trap minimum.

The coherent states require an infinite number of terms to represent analytically, in either eigenfunction or coordinate representation.

Quantum Tunneling

This wavefunction has an expectation energy smaller than the height of the potential.

In a classical analog, this would mean the particle cannot be found beyond the potential barrier. However, when we evolve the wavefunction through time...

we find a non-negligible wavefunction amplitude can emerge beyond the barrier, after exponentially decaying within. The wavefunction has partially reflected off of, and partially transmitted through, the barrier.

This implies a possibility of measuring the particle beyond the classically insurmountable barrier.

We say the particle has tunneled through the barrier.