Quantum Harmonic Oscillator

1. Quantum Harmonic Oscillator Quantum Harmonic Oscillator 2006 Quantum Mechanics Prof. Y. F. Chen

2. Quantum Harmonic Oscillator Quantum Harmonic Oscillator • 1D S.H.O.：linear restoring force , k is the force constant & parabolic potential . • harmonic potential’s minimum at = a point of stability in a system A particle oscillating in a harmonic potential 2006 Quantum Mechanics Prof. Y. F. Chen

3. Quantum Harmonic Oscillator Quantum Harmonic Oscillator • Ex：the positions of atoms that form a crystal are stabilized by the presence of a potential that has a local min at the location of each atom → ∵ the atom position is stabilized by the potential, a local min results in the first derivative of the series expansion = 0 ∴ • → a local min in V(x) is only approximated by the quadratic function of a H.O. 2006 Quantum Mechanics Prof. Y. F. Chen

4. Quantum Harmonic Oscillator Schrödinger Wave Eq. for 1D Harmonic Oscillator • for the H.O. potential, the time-indep Schrödinger wave eq.： use(1) & (2) → making the substitution → called Hermite functions. 2006 Quantum Mechanics Prof. Y. F. Chen

5. Quantum Harmonic Oscillator Hermite Functions • One important class of orthogonal polynomials encountered in QM & laser physics is the Hermite polynomials, which can be defined by the formula the first few Hermite polynomials are： in general： 　　 . 2006 Quantum Mechanics Prof. Y. F. Chen

6. Quantum Harmonic Oscillator Hermite Functions • the Hermite polynomials come from the generating function： . → Taylor series： . → • substituting into 　 ： → recurrence relation： 2006 Quantum Mechanics Prof. Y. F. Chen

7. Quantum Harmonic Oscillator Hermite Functions • substituting into 　 ： → recurrence relation： • with & → 2nd-order ordinary differential equation for • eigenvalues of the 1D quantum H.O.： 2006 Quantum Mechanics Prof. Y. F. Chen

8. Quantum Harmonic Oscillator Stationary States of 1D Harmonic Oscillator • the eigenfunctions of 1D H.O.： with the help of , find normalization constant , → • (i) in CM, the oscillator is forbidden to go beyond the potential, beyond the turning points where its kinetic energy turns negative. (ii) the quantum wave functions extend beyond the potential, and thus there is a finite probability for the oscillator to be found in a classically forbidden region 2006 Quantum Mechanics Prof. Y. F. Chen

9. Quantum Harmonic Oscillator Stationary States of 1D Harmonic Oscillator 2006 Quantum Mechanics Prof. Y. F. Chen

10. Quantum Harmonic Oscillator Stationary States of 1D Harmonic Oscillator • the classical probability of finding the particle inside a region　　 ： . the velocity can be expressed as a function of 　： → 2006 Quantum Mechanics Prof. Y. F. Chen

11. Quantum Harmonic Oscillator Stationary States of 1D Harmonic Oscillator • (i) the difference between the two probabilities for n=0 is extremely striking ∵there is no zero-point energy in CM (ii) the quantum and classical probability distributions coincide when the quantum number n becomes large (iii) this is an evidence of Bohr’s correspondence principle 2006 Quantum Mechanics Prof. Y. F. Chen

12. Quantum Harmonic Oscillator Stationary States of 1D Harmonic Oscillator • (1) classically, the motion of the H.O. is in such a manner that the position of the particle changes from one moment to another. (2) however, although there is a probability distribution for any eigenstate in QM, this distribution is indep of time → stationary states (3) even so, the Ehrenfest theorem reveals that a coherent superposition of a number of eigenstates, i.e., so-called “wave packet state”, will lead to the classical behavior 2006 Quantum Mechanics Prof. Y. F. Chen

13. Quantum Harmonic Oscillator Stationary States of 1D Harmonic Oscillator • show ： using the generation function , we can have ∵ the orthogonality property, the integration leads to → as a consequence, we can obtain 2006 Quantum Mechanics Prof. Y. F. Chen

14. Quantum Harmonic Oscillator The Poisson Distribution • given a mean rate of occurrence r of the events in the relevant interval, the Poisson distribution gives the probability that exactly n events will occur • for a small time interval the probability of receiving a call is . the probability of receiving no call during the same tiny interval is given by . the probability of receiving exactly n calls in the total interval is given by 2006 Quantum Mechanics Prof. Y. F. Chen

15. Quantum Harmonic Oscillator The Poisson Distribution • rearranging , dividing through by , and letting , the differential recurrence eq. can be found and written as for ： which can be integrated to lead to with the fact that the probability of receiving no calls in a zero time interval must be equal to unity： 2006 Quantum Mechanics Prof. Y. F. Chen

16. Quantum Harmonic Oscillator The Poisson Distribution • substituting into for ： , repeating this process, can be found to be • the sum of the probabilities is unity： • the mean of the Poisson distribution： 　 2006 Quantum Mechanics Prof. Y. F. Chen

17. Quantum Harmonic Oscillator The Poisson Distribution • in other words, the Poisson distribution with a mean of is given by： 2006 Quantum Mechanics Prof. Y. F. Chen

18. Quantum Harmonic Oscillator Schrödinger Coherent States of the 1D H.O. • The Schrödinger coherent wave packet state can be generalized as with • it can be found that the norm square of the coefficient is exactly the same as the Poisson distribution with the mean of 2006 Quantum Mechanics Prof. Y. F. Chen

19. Quantum Harmonic Oscillator Schrödinger Coherent States of the 1D H.O. • substituting & into using 2006 Quantum Mechanics Prof. Y. F. Chen

20. Quantum Harmonic Oscillator Schrödinger Coherent States of the 1D H.O. • as a result, the probability distribution of the coherent state is given by： • it can be clearly seen that the center of the wave packet moves in the path of the classical motion 2006 Quantum Mechanics Prof. Y. F. Chen

21. Quantum Harmonic Oscillator Creation & Annihilation Operators • with , & the operator acting on the eigenstate 2006 Quantum Mechanics Prof. Y. F. Chen

22. Quantum Harmonic Oscillator Creation & Annihilation Operators • in a similar way, the operator acting on the eigenstate 2006 Quantum Mechanics Prof. Y. F. Chen

23. Quantum Harmonic Oscillator Creation & Annihilation Operators • → & • consequently, it is convenient to define 2 new operators： & 2006 Quantum Mechanics Prof. Y. F. Chen

24. Quantum Harmonic Oscillator Creation & Annihilation Operators • the operator is the increasing (creation) operator： this means that operating with on the n-th stationary states yields a state, which is proportional to the higher (n +1)-th state • the operator is the lowering (annihilation) operator： this means that operating with on the n-th stationary states yields a state, which is proportional to the higher (n -1)-th state 2006 Quantum Mechanics Prof. Y. F. Chen

25. Quantum Harmonic Oscillator Creation & Annihilation Operators • in terms of & , the operators & can be expressed as： & • we can find the commutator of these 2 ladder operators： which is the so-called canonical commutation relation 2006 Quantum Mechanics Prof. Y. F. Chen

26. Quantum Harmonic Oscillator Creation & Annihilation Operators • is the hermitian conjugate　： proof： 2006 Quantum Mechanics Prof. Y. F. Chen

27. Quantum Harmonic Oscillator Creation & Annihilation Operators • with , & the operator acting on the eigenstate 2006 Quantum Mechanics Prof. Y. F. Chen

28. Quantum Harmonic Oscillator Creation & Annihilation Operators • in a similar way, the operator acting on the eigenstate 2006 Quantum Mechanics Prof. Y. F. Chen

29. Quantum Harmonic Oscillator Creation & Annihilation Operators • → & • consequently, it is convenient to define 2 new operators： & 2006 Quantum Mechanics Prof. Y. F. Chen

30. Quantum Harmonic Oscillator Creation & Annihilation Operators • the operator is the increasing (creation) operator： this means that operating with on the n-th stationary states yields a state, which is proportional to the higher (n +1)-th state • the operator is the lowering (annihilation) operator： this means that operating with on the n-th stationary states yields a state, which is proportional to the higher (n -1)-th state 2006 Quantum Mechanics Prof. Y. F. Chen

31. Quantum Harmonic Oscillator Creation & Annihilation Operators • in terms of & , the operators & can be expressed as： & • we can find the commutator of these 2 ladder operators： which is the so-called canonical commutation relation 2006 Quantum Mechanics Prof. Y. F. Chen

32. Quantum Harmonic Oscillator Creation & Annihilation Operators • is the hermitian conjugate　： proof： 2006 Quantum Mechanics Prof. Y. F. Chen

33. Quantum Harmonic Oscillator Creation & Annihilation Operators • with & → using the commutation relation → • define the so-called number operator： →the H.O. Hamiltonian takes the form： 2006 Quantum Mechanics Prof. Y. F. Chen

34. Quantum Harmonic Oscillator Creation & Annihilation Operators • the eigenstates of can be found to be coherent states ： • coherent states have the minimum uncertainty 2006 Quantum Mechanics Prof. Y. F. Chen

35. Quantum Harmonic Oscillator Creation & Annihilation Operators as a consequence, we obtain the minimum uncertainty state： 2006 Quantum Mechanics Prof. Y. F. Chen