1. PHYS 773: Quantum Mechanics February 6th, 2012
Algebraic Method of Solving the Linear Harmonic Oscillator Lecture by Gable Rhodes
Refer to Merzbacher, p and Griffiths, p40-51
PHYS 773: Quantum Mechanics
February 6th, 2012

2. Simple Harmonic Oscillator
Many physical problems can be modeled as small oscillations around a stable equilibrium
Potential is described as parabolic around the minimum energy
The Hamiltonian is formulated in the usual way for canonical variables q and p.
Gable Rhodes, February 6th, 2012

3. Raising & Lowering Operators - Defined
After substituting in our potential, we can rearrange the constants slightly to give the Hamiltonian a more suggestive appearance
If we “factor” the operators, we get
Recognizing that the last term is the commutator of the canonical variables, we get
Gable Rhodes, February 6th, 2012

4. Raising & Lowering Operators - Defined
We can now define the operator a
Upon substitution
And Simplifying
Gable Rhodes, February 6th, 2012

5. Raising & Lowering Operators - Defined
If we reverse the order of the operators, a similar expression is obtained
Subtracting the two forms yields the commutator
Simplifying to
Gable Rhodes, February 6th, 2012

6. Raising & Lowering Operators - Defined
It is important to note that the a, a† operators are defined in such a way that as long as the state variables follow the canonical commutation relation, the a, a† commutator will be 1.
And the Hamiltonian can be written as a linear function of a, a†
Gable Rhodes, February 6th, 2012

7. Properties of a, a†
If a wave function has the property that it is an eigenfunction of the Hamiltonian
We can introduce the equivalent statement for the operator aa† with generic eigenvalue, λ
We then apply the a† operator to a†ψ and test if the result is the same eigenvector.
Use commutator
Gable Rhodes, February 6th, 2012

8. Properties of a, a† And the equivalent method for a
Using the relationship of Hamiltonian, we can then relate eigenvalues
Gable Rhodes, February 6th, 2012

9. Raising & Lowering Operators -Properties
a† is called the raising (or creation) operator.
And a is the lowering (or annihilating) operator.
The rungs of the ladder are all evenly separated.
No degeneracy.
a† a† ψ
λ+2
a† ψ
λ+ 1 ψ λ aψ
λ- 1
aaψ
λ-2
Wave vectors
Eigenvalues
Gable Rhodes, February 6th, 2012

10. What is the Significance?
If we have any solution, we can find infinitely more solutions by repeated application of the raising and lowering operators
But, importantly, although there are infinite solutions we know that the are are no solutions with negative energy (both kinetic and potential components for the Hamiltonian are always positive)
Therefore, a ground state must exist. (this is also a property of Sturm-Liouville PDE)
Applying the lowering operator to the ground state will result in a null vector (trivial state).
Eq.10-77
Gable Rhodes, February 6th, 2012

11. Raising & Lowering Operators -Eigenvectors
Once we have a ground state, repeated application of the raising operator will result in an infinite set of eigenvectors with distinct (non-degenerate) eigenvalues.
And introducing an arbitrary starting point λ0.
But for the special case of the ground state
Must be zero on the right side (by our definition), so λ0 is exactly 0.
Gable Rhodes, February 6th, 2012

12. Determine the Energy Levels
Using the previous equation relating the Hamiltonian and aa†, we can relate energy to λ (n).
And since we started at the ground state, we can relate the energy level to the eigenvectors
If we use a normalization constant
Where An can be found by direct integration at each step or by algebraic tricks (later)
Gable Rhodes, February 6th, 2012

13. What Are These Operators Good For Anyway?
We found Energy exactly and wavevectors in abstract form.
What else can we do with them?
What about expectation values? In chapter 5, problem 2, we were asked to find <V>. This required direct integration with the (explicitly known) wavefunction.
Can this be done without knowing the wavefunction?
Gable Rhodes, February 6th, 2012

14. Expectation value of Potential
If we use the definition of a†, a and rewrite q and p operators in terms of a and a† we get
Now these can be substituted into <V>
Gable Rhodes, February 6th, 2012

15. Expectation value of Potential
Of the four terms in the integral, we see that two of them vanish due to the orthogonality of the wavevectors.
And the other two are known to us from the previous work.
Gable Rhodes, February 6th, 2012

16. Expectation value of Potential
This makes the result pretty straightforward
And the this result agrees with problem 5.2, and the virial theorem
But, we did not need to know the explicit form of the wavefunction.
Gable Rhodes, February 6th, 2012

17. <q>, <p>, <q2>, <p2>
Other key expectation values can be easily obtained.
Gable Rhodes, February 6th, 2012

18. Variance and Uncertainty?
The variance is therefore
And this agrees with the uncertainty principle
Gable Rhodes, February 6th, 2012

19. What is left then?
The Normalization constant, which can also be determined algebraically.
Square both sides and integrate
After integration by parts and throwing out the boundary terms.
So that gives us our wavefunction in terms of the ground state and raising operator
Gable Rhodes, February 6th, 2012

20. Normalization Lets try ψ1. Next try ψ2.
We can now write the general equation
Gable Rhodes, February 6th, 2012

21. And what is ψ0? Starting with our condition for the ground state
And using the definition of the operator
We get a first order ODE
Gable Rhodes, February 6th, 2012

22. And what is ψ0? The equation is separable and easily solved.
After normalization we get
Which is consistent with the solutions in chapter 5.
Gable Rhodes, February 6th, 2012

23. Finding ψ1.
Here we can use the raising operator to generate further solutions
Substitute in the operator
After rearranging, we get the desired result.
Gable Rhodes, February 6th, 2012

24. Finding ψn.
We find that continuing up the ladder is in fact a generating algorithm for the Hermite polynomials, and the general equation then identical to eq. 5.39
Gable Rhodes, February 6th, 2012

25. Matrix representation of a, a†
We can show that the lowering operator relates neighboring wave vectors with the normalization factor.
Adding the bra.
This gives the matrix elements of a as shown.
Gable Rhodes, February 6th, 2012

26. Matrix representation of a, a†
As an example, lowering n=2,
Gable Rhodes, February 6th, 2012

27. Matrix representation of a, a†
The equivalent representation of the raising operator is derived from the expression
Adding the bra.
Gable Rhodes, February 6th, 2012

28. Matrix representation of a, a†
With the lowering and raising operators in matrix form, we can then solve for the q and p operators in terms of a, a†.
For position we get,
Gable Rhodes, February 6th, 2012

29. Matrix representation of a, a†
And the equivalent expression for momentum is,
The complex constant insures that the anti-symmetric matrix is Hermitian.
Gable Rhodes, February 6th, 2012