Algebraic Method of Solving the Linear Harmonic Oscillator Lecture by Gable Rhodes

1. Algebraic Method of Solving the Linear Harmonic OscillatorLecture by Gable Rhodes 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 propertythat 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. λ+2 a† a† ψ λ+1 a† ψ 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 Ancan 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 operatoris 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