1. 5. 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’(x 0 ) = 0 Constant term is irrelevant We can arbitrarily choose the minimum to be x 0 = 0 We define the classical angular frequency  so that All Problems are the Harmonic Oscillator

2. Raising and Lowering Operators 5A. The 1D Harmonic Oscillator 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 the proportionality constants:

5. 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 What are the possible eigenvalues

6. Sometimes – rarely – we want the wave functions Let’s see if we can find the ground state |0  : The Wave Functions (1) 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. Working with the Harmonic Oscillator 5B. Working with the H.O. & Coherent States 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. All Problems are the Harmonic Oscillator 5C. Multiple Particles and 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 X 0. Recall derivative vanishes at minimum A constant term in the Hamiltonian never matters We can always change origin to X 0 = 0. Now define: We now have:

15. Solving if it’s Diagonal To simplify, assume k ij has only diagonal elements: We define  i 2 = k i /m: Next define Find the commutators: Write the Hamiltonian in terms of these: Eigenstates and Eigenenergies:

16. Note that the matrix made of k ij ’s is a real symmetric matrix (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 new coordinates, the behavior is much simpler. The matrix V diagaonalizes K Will this approach work quantum mechanically? What if it’s Not Diagonal?

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

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

19. Sample Problem Name the eigenstates and find the corresponding energies of the Hamiltonian Find the coefficients k ij that make up the K matrix NO! Remember, k ij must be symmetric! So k 12 = k 21 Now find the eigenvalues: The states and energies are:

20. It Isn’t Really That Complex 5C. The Complex Harmonic Oscillator 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 everything about quantizing this: More usefully, write them in terms of raising and lowering operators: 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 their derivatives, we substitute: This is exactly what we will need when we quantize EM fields later