Harmonic oscillator and coherent states

1. Harmonic oscillator and coherent states • Energy eigen states by algebra method • Wavefunction • Coherent state • The most classical quantum system • Reading materials: • Chapter 7 of Shankar’s PQM.

2. Algebra method for eigen states The Hamiltonian of a harmonic oscillator is Defining dimensionless coordinate position and momentum are now on equal foot. Classical dynamics: Cyclic trajectory in the phase space.

3. Quantum mechanics: Let us “rotate” the coordinates by an imaginary angle so that the cyclic rotation in the phase space is automatically taken into account by the transformation Note: Do you still remember how to write a circularly polarized light (whose electric field rotates) in terms of linear polarization? The Hamiltonian becomes How to get the eigen states and the eigen values?

4. A few general properties to be used: 1. The energy spectrum must be lower bounded, for So there must be a ground state 2. There should be no continuum state, i.e., the eigenstate wavefunction should be normalized to one. Otherwise the energy of the state cannot be finite due to the infinite potential diverging at the remote positions. 3. Theorem: In one-dimension space, a discrete state cannot be degenerate (see Shankar, PQM page 176 for a proof).

5. If there is an eigenstate Repeating the process, we get a series of eigen states Their energies form an equally space ladder

6. The energy ladder has to be lower bounded, so we must have So the ground state energy is given by All the other eigen states can be obtained by

7. matrix forms in the eigen state basis:

8. The dimensionless position and momentum operators are matrix forms in the eigen state basis:

9. Wavefunctions Let us first consider the ground state: In the dimensional coordinates: The solution is A nice property of Gaussian function is that its Fourier transformation is also a Gaussian function. The wavefunction in the q-representation would have the same form (remember Y and Q are inter-exchangeable. So the ground state wavefunction in the real space is: A wavepacket centered at the potential minimum.

10. Now consider the excited states: Thus the wavefunction in real space is The above equation actually defines the generation of Hermite polynomials. The wavefunction in the momentum-representation is defining a nice property of the F.T. of Hermite polynomials.

11. Regarded as boson Now that all the eigen states of a harmonic oscillator are equally spaced., we can take the rising from one state to the next one as the addition of one particle with the same energy to a mode. The ground state contains no particle and hence is the vacuum state. Simple one mode can have an arbitrary number of particles, this particle is a boson. Vacuum state The state with n bosons, called a Fock state The operator annihilating one boson The operator creating one boson The boson particle number operator The energy of the boson The energy of the vacuum

12. Coherent state A coherent state of a harmonic oscillator is defined as In terms of the position and momentum operators, it is

13. Coherent state in Fock state basis To derive the wave function of the coherent state in the Fock state basis, we use the Baker-Hausdorff theorem so the condition for the theorem is satisfied.

14. Coherent state in Fock state basis The expansion is

15. The state is normalized (of course) as can be checked directly: Coherent state is an eigen state of the annihilation operator: Removing one boson does NOT change a coherent state! However, adding one boson changes the state

16. The expectation value of the boson number is Boson number distribution (Poisson distribution)

17. To understand the nature of the coherent state, let us consider first a few special case: The real space wave function of this state is Or in the momentum representation The state is the ground state with the momentum distribution shifted by

18. The wave function in the momentum representation is The real space wave function of this state is The state is the ground state with the real-space distribution shifted by

19. Baker-Hausdorff theorem For more, do the homework.

20. Quantum fluctuation According to Heisenberg principle Fock states

21. In particular, for the vacuum state The vacuum state has minimum quantum fluctuation (the most classical state)

22. Quantum fluctuation: Coherent state For a coherent state The center of the wavepacket in the phase space is at

23. The variance can be obtained by calculating A coherent state has minimum quantum fluctuation. This justify our viewing a coherent state as a wavepacket centered at a point in the phase space and a most classical state.

24. Coherent states as a basis The coherent states for all complex numbers form a complete basis Completeness condition:

25. The Hilbert space has been expanded by a discrete set of states (the Fock states). But the complex numbers form a continuum. So the coherent states must be over complete, i.e., more than enough, since they are not orthogonal:

26. The most classical quantum system

27. We have seen that a coherent state can be viewed as a shift in the phase space from the vacuum state. The vacuum state, of course, is also a coherent state. What if we do the shift from a coherent state other than the vacuum? Repeatedly using Baker-Hausdorff theorem: It is still a coherent state, up to a trivial phase factor.

28. A shift in the phase space from a coherent state is still a coherent state, the total shift from the vacuum is just the sum of the two shifts. Thus we can define a shift operator

29. Time evolution of a coherent state If we have an initial coherent state: The time evolution is simply It is a coherent state with its shift from the vacuum rotating in the phase space like a classical oscillator.

30. Harmonic oscillator driven by a force Let us consider the motion of a harmonic oscillator starting from the ground state In the form of boson operators: Suppose the state at a certain time is The Schroedinger equation is Consider an infinitesimal time increase

31. The first term is a free evolution of the state. The second term is the shift operator which shift a state in the phase space along the position axis. The evolution from the initial state in a finite time is

32. So, if the initial is a coherent state, say, the vacuum state, the state after a finite time of evolution is still a coherent state And we have the shift for an infinitesimal time increase to be The equation of motion is: The same as the classical eqns. for position and momentum! i.e.,

33. Conclusion: A harmonic oscillator driven by a classical force from the ground state is always in a coherent state. We have seen that the coherent state follows basically the equations for the classical eqns for position and momentum. It could be taken as a reproduction of the classical dynamics from quantum mechanics. The coherent state could be understood as classical particle, though it is quite a wavepacket (it is just so small that we had not enough resolution to tell it from a particle). So, if we have only classical forces and harmonic oscillators, there is no way to obtain a “real” quantum state from the vacuum or the ground state. That is why we call harmonic oscillators the most classical quantum systems.