5 The Harmonic Oscillator All Problems are the
5. The Harmonic Oscillator All Problems are the Harmonic Oscillator Consider a general problem in 1 D • 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
5 A. The 1 D 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:
Commutators and the Hamiltonian • We will need the commutator • Now let’s work on the Hamiltonian
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:
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
The Wave Functions (1) • Sometimes – rarely – we want the wave functions • Let’s see if we can find the ground state |0 : • Normalize it:
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
5 B. 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: Abandon all hope all ye who enter here • The right way to do this:
Sample Problem At t = 0, a 1 D harmonic oscillator system is in the state (a) Find the quantum state at arbitrary time (b) Find P at arbitrary time
Sample Problem (2) (b) Find P at arbitrary time
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
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
Sample Problem Find X for the coherent state |z
5 C. 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 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:
Solving if it’s Diagonal • To simplify, assume kij has only diagonal elements: • We define i 2 = ki/m: • Next define • Find the commutators: • Write the Hamiltonian in terms of these: • Eigenstates and Eigenenergies:
What if it’s Not Diagonal? • Note that the matrix made of kij’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?
Does this Work Quantum Mechanically? • 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:
The Hamiltonian Rewritten: The procedure: • Find the eigenvectors |v and eigenvalues ki 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!
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 k 12 = k 21 • Now find the eigenvalues: The states and energies are:
5 C. 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 everything about quantizing this: • More usefully, write them in terms of raising and lowering operators: • The Hamiltonian is now:
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:
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
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