NDEigensystem

Mathematics
Physics

Schrödinger Equation for the Linear Harmonic Oscillator

Find the first few eigenvalues and eigenfunctions

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The Schrödinger equation for the linear harmonic oscillator is given by:

ℏ

ψ"(x)

ψ(x)=ϵψ(x)

For simplicity scale the variables such that

ℏ=m=k=1

In[1]:=

ℒ=-

ψ''[x]+

ψ[x];

Use the

NDEigensystem

algorithm to get the first five eigenvalues and eigenfunctions:

In[2]:=

{evals,efns}=NDEigensystem[ℒ,ψ[x],{x,-5,5},5];

List the eigenvalues:

In[3]:=

evals

Out[3]=

{0.500079,1.50054,2.5019,3.5047,4.50941}

The exact values are

ϵ=

, … Visualize the eigenvalues by plotting them on the same scale as the potential energy

V(x)=

In[4]:=

Plot

2,Table[evals〚n〛,{n,1,5}],{x,-3.1,3.1}

Out[4]=

Plot the first four eigenfunctions:

In[5]:=

Table[Plot[Evaluate[-efns〚n〛],{x,-3,3},GridLinesAutomatic],{n,1,4}]

Out[5]=





We can solve the Schrödinger equation exactly for the ground state,

ϵ=

, subject to the condition that

ψ(x)0

x

±∞:

In[6]:=

DSolve-

ψ''[x]+

ψ[x]

ψ[x],ψ[∞]0,ψ[-∞]0,ψ[x],x//Quiet

Out[6]=

ψ[x]







For the first excited state, with

ϵ=

In[7]:=

DSolve-

ψ''[x]+

ψ[x]

ψ[x],ψ[∞]0,ψ[-∞]0,ψ[x],x//Quiet

Out[7]=

ψ[x]



ParabolicCylinderD1,

x

To get a more elementary form:

In[8]:=

SeriesParabolicCylinderD1,

x,{x,∞,4}//Normal

Out[8]=



Related Symbols

NDEigensystem

Publisher Information

Contributed by: S. M. Blinder