5. The Harmonic Oscillator

1. 5. The Harmonic Oscillator All Problems are the Harmonic Oscillator Consider a general problem in 1D • Particles tend to be near their minimum • Taylor expand V(x) near its minimum • Recall V’(x0) = 0 • Constant term is irrelevant • We can arbitrarily choose the minimum to be x0 = 0 • We define the classical angular frequency  so that

2. 5A. The 1D Harmonic Oscillator Raising and Lowering Operators • First note that V() = , so only bound states • Classically, easy to show that the combination mx + ip has simple behavior • With a bit of anticipation, we define • We can write X and P in terms of these:

3. Commutators and the Hamiltonian • We will need the commutator • Now let’s work on the Hamiltonian

4. Raising and Lowering the Eigenstates • Let’s label orthonormal eigenstates by their a†a eigenvalue If we act on an eigenstate with a or a†, it is still an eigenstate of a†a: • Lowering Operator: • Raising Operator: • We can work out theproportionality constants:

5. What are the possible eigenvalues • It is easy to see that since ||a|n||2 = n, we must have n  0. • This seems surprising, since we can lower the eigenvalue indefinitely • This must fail eventually, since we can’t go below n = 0 • Flaw in our reasoning: we assumed implicitly that a|n  0 • If we lower enough times, we must have a|n = 0  ||a|n||2 = 0 • Conclusion: if we lower n repeatedly, we must end at n = 0 • n is a non-negative integer • If we have the state |0, we can get other states by acting with a† • Note: |0  0

6. The Wave Functions (1) • Sometimes – rarely – we want the wave functions • Let’s see if we can find the ground state |0: • Normalize it:

7. The Wave Functions (2) • Now that we have the ground state, we can get the rest • Almost never use this! • If you’re doing it this way, you’re doing it wrong n = 3 n = 2 n = 1 n = 0

8. 5B. Working with the H.O. & Coherent States Working with the Harmonic Oscillator • It is common that we need to work out things like n|X|m or n|P|m • The wrong way to do this: • The right way to do this: Abandon all hope all ye who enter here

9. Sample Problem At t = 0, a 1D harmonic oscillator system is in the state (a) Find the quantum state at arbitrary time (b) Find P at arbitrary time

10. Sample Problem (2) (b) Find P at arbitrary time

11. Coherent States Can we find eigenstates of a and a†? • Yes for a and no for a† • Because a is not Hermitian, they can have complex eigenvalues z • Note that the state |z = 1 is different from |n = 1 • Let’s find these states: • Act on both sides with m|: • Normalize it

12. Comments on Coherent States They have a simple time evolution • Suppose at t = 0, the state is • Then at t it will be When working with this state, avoid using the explicit form • Instead use: • And its Hermitian conjugate equation: • Recall: these states are eigenstates of a non-Hermitian operator • Their eigenvalues are complex and they are not orthogonal • These states roughly resemble classical behavior for large z • They can have large values of X and P • While having small uncertainties X and P

13. Sample Problem Find X for the coherent state |z

14. 5C. Multiple Particles and Harmonic Oscillator All Problems are the Harmonic Oscillator • Consider N particles with identical mass m in one dimension • This could actually be one particle in N dimensions instead • These momenta & position operators have commutation relations: • Taylor expand about the minimum X0. Recall derivative vanishes at minimum • A constant term in the Hamiltonian never matters • We can always change origin to X0 = 0. • Now define: • We now have:

15. Solving if it’s Diagonal • To simplify, assume kijhas only diagonal elements: • We define i2 = ki/m: • Next define • Find the commutators: • Write the Hamiltonian in terms of these: • Eigenstates and Eigenenergies:

16. What if it’s Not Diagonal? • Note that the matrix made of kij’s is a real symmetricmatrix (Hermitian) Classically, we would solve this problem by finding the normal modes • First find eigenvectors of K: • Since K is real, these are real eigenvectors • Put them together into a real orthogonal matrix • Same thing as unitary, but for real matrices • Then you can change coordinates: • Written in terms of the newcoordinates, the behavior is much simpler. • The matrix V diagaonalizesK • Will this approach work quantum mechanically?

17. Does this Work Quantum Mechanically? • Define new position and momentum operators as • Because V is orthogonal, these relationsare easy to reverse • The commutation relations for these are: • We now convert this Hamiltonian to the new basis:

18. The Hamiltonian Rewritten: The procedure: • Find the eigenvectors |v and eigenvalueski of the K matrix • Use these to construct V matrix • Define new operators Xi’ and Pi’ • The eigenstates and energies are then: Comments: • To name states and find energies, all you need is eigenvalues ki • Don’t forget to write K in a symmetric way!

19. Sample Problem Name the eigenstates and find the corresponding energies of the Hamiltonian • Find the coefficients kij that make up the K matrix • NO! Remember, kij must be symmetric! So k12 = k21 • Now find the eigenvalues: The states and energies are:

20. 5C. The Complex Harmonic Oscillator It Isn’t Really That Complex • A classical complex harmonic oscillator is a system with energy given by Where z is a complex position • Just think of z as a combination of two real variables: • Substituting this in, we have: • We already know everythingabout quantizing this: • More usefully, write them interms of raising and loweringoperators: • The Hamiltonian is now:

21. Working with complex operators • Writing z in terms of a and a† • Let’s define for this purpose • Commutation relations: • All other commutators vanish • In terms of these, • And the Hamiltonian:

22. The Bottom Line • If we have a classical equation for the energy: • Introduce raising/lowering operators with commutation relations • The Hamiltonian in terms of these is: • Eigenstates look like: • For z and z* and theirderivatives, we substitute: • This is exactly what we will need when we quantize EM fields later