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

2. 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
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. Schrödinger Wave Eq. for 1D Harmonic Oscillator
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. Hermite Functions substituting into ： → recurrence relation： with &
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. Stationary States of 1D Harmonic Oscillator
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. Stationary States of 1D Harmonic Oscillator
Quantum Harmonic Oscillator
Stationary States of 1D Harmonic Oscillator
2006 Quantum Mechanics
Prof. Y. F. Chen

10. Stationary States of 1D Harmonic Oscillator
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. Stationary States of 1D Harmonic Oscillator
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. Stationary States of 1D Harmonic Oscillator
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. Stationary States of 1D Harmonic Oscillator
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. The Poisson Distribution
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. The Poisson Distribution
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. The Poisson Distribution
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. The Poisson Distribution
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. Schrödinger Coherent States of the 1D H.O.
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. Schrödinger Coherent States of the 1D H.O.
Quantum Harmonic Oscillator
Schrödinger Coherent States of the 1D H.O.
substituting & into
using
2006 Quantum Mechanics
Prof. Y. F. Chen

20. Schrödinger Coherent States of the 1D H.O.
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. Creation & Annihilation Operators
Quantum Harmonic Oscillator
Creation & Annihilation Operators
with , &
the operator acting on the eigenstate
2006 Quantum Mechanics
Prof. Y. F. Chen

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

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

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

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

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

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

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

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