The 1-D Harmonic Oscillator

Consider a particle of mass m moving in a potential

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The force on the particle is given by

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The motion of the classical particle is governed by Newton’s second law

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The general solution to this equation is

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with

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We have

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The total energy is independent of time, we have a conservative system.

To make the transition to Quantum Mechanics we express E in terms of p and x and then replace the classical quantities p and x by their corresponding quantum mechanical operators P and X.

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Finding the eigenvalues of H

Define

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where

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  have the same eigenstates, their eigenvalues differ by a factor of .

 do not commute.

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Let us define some new operators and investigate their properties. Define

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Then

 

Note that a and aT do not commute.

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Now we study the product operator

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Similarly,

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We may therefore write

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Define the number operator N=aTa.  Then .  and N have common eigenstates, finding the eigenstates of N is equivalent to finding the eigenstates of .  The operator N does not commute with a and aT.

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We will now study some of the properties of the eigenstates of N and therefore the eigenstates of .

(a) The eigenvalues of N are positive.
Proof: Let |fn> be an eigenstate of N with eigenvalue n. 
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Find the norm of the vector a|fn>. , 
since the norm is always positive, and .
(b) If |fn> is an eigenstate of N with eigenvalue and then a|fn> is an eigenstate of N with eigenvalue n-1.  If |fn> is an eigenstate of N with eigenvalue then n=0 and .
Proof: , unless n=0.
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is non zero unless n=0.  Therefore is an eigenvector of N with eigenvalue n-1.  But the eigenvalues have to be positive.  Therefore if n<1 we need =0 which requires n=0.
(c) If |fn> is an eigenstate of N with eigenvalue and then aT|fn> is an eigenstate of N with eigenvalue n+1.
Proof: .
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(d) The eigenvalues of N are the non negative integers.
Proof: N has at least one non-zero eigenvector with eigenvalue n.  
(It is a Hermitian operator).  
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Then is also an eigenvector if (n-1)>0, 
is also an eigenvector if (n-n)>0.  
But for some n’=n+1 n-n’ will become smaller than zero.  
Therefore must be zero.  
But is only zero if or (n-n)=0.  
This means that n is a non-negative integer.

If |fn> is a non zero eigenvector of N with integer eigenvalue n, then the |fn’> , n’=0,1,2,... are also eigenvectors.  (We can generate these eigenvectors by applying a or aT to |fn> . )

The eigenvectors of N are {|fn> } with eigenvalues {n}, where n is a non-negative integer.

The eigenvectors of are {|fn> } with eigenvalues {n+}.

The eigenvectors of H are {|fn> } with eigenvalues {}.

The energy of the harmonic oscillator is quantized.  The ground state energy is . The energy levels are evenly spaced.

(e) The ground state of is not degenerate.
Proof: .

.  In the {|x>} representation this becomes

,

 is the solution to this first order differential equation. It is unique up to a multiplicative constant C.  Therefore there exists only one normalized f0(x) and |f0> with the eigenvalue 0.  The eigenstate corresponding to the lowest eigenvalue of is not degenerate.

By induction we can now proof that all eigenstates of  are not degenerate.

The eigenvectors {|fn> } of are a basis for the state space Ex of a particle in a one dimensional problem.  The |fn> are orthogonal since all eigenvalues are not degenerate. We now want to choose a normalized set {|fn> }.

Assume |f0> is normalized, <f0|f0>=1.  
The normalized vector |f1> is |f1>=c1aT|f0>. 
If <f1|f1>=1, then |c1|2 <f0|aaT|f0>=|c1|2 <f0|aTa+1|f0>=|c1|2 =1. 
Therefore |f1>=aT|f0>.
The normalized vector |fn> is |fn>=cnaT|fn-1>.
If <fn|fn>=1, then |cn|2 <fn-1|aaT|fn-1>=|cn|2<fn-1|aTa+1|fn-1>=n|cn|2 =1.  
Therefore |fn>=aT|fn-1>.

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The set forms an orthonormal basis for Ex .

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Let {|fn>} denote this orthonormal eigenbasis of the operator .

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The results of operating with a or aT on |fn> are given by

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The matrices representing a and aT in the {|fn>} basis therefore are

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The order of the basis vectors is

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In terms of a and aT the observables X an P are given by

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The matrices representing X and P therefore are

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X and P are Hermitian operators.