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Quantum Harmonic Oscillator 2006 Quantum MechanicsProf. Y. F. Chen Quantum Harmonic Oscillator.

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Presentation on theme: "Quantum Harmonic Oscillator 2006 Quantum MechanicsProf. Y. F. Chen Quantum Harmonic Oscillator."— Presentation transcript:

1 Quantum Harmonic Oscillator 2006 Quantum MechanicsProf. Y. F. Chen Quantum Harmonic Oscillator

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

3 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 Quantum MechanicsProf. Y. F. Chen Quantum Harmonic Oscillator

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

5 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 Quantum MechanicsProf. Y. F. Chen Hermite Functions Quantum Harmonic Oscillator

6 the Hermite polynomials come from the generating function. Taylor series. substituting into recurrence relation 2006 Quantum MechanicsProf. Y. F. Chen Hermite Functions Quantum Harmonic Oscillator

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

8 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 MechanicsProf. Y. F. Chen Stationary States of 1D Harmonic Oscillator Quantum Harmonic Oscillator

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

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

11 (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 Bohrs correspondence principle 2006 Quantum MechanicsProf. Y. F. Chen Stationary States of 1D Harmonic Oscillator Quantum Harmonic Oscillator

12 (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 MechanicsProf. Y. F. Chen Stationary States of 1D Harmonic Oscillator Quantum Harmonic Oscillator

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

14 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 MechanicsProf. Y. F. Chen The Poisson Distribution Quantum Harmonic Oscillator

15 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 MechanicsProf. Y. F. Chen The Poisson Distribution Quantum Harmonic Oscillator

16 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 MechanicsProf. Y. F. Chen The Poisson Distribution Quantum Harmonic Oscillator

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

18 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 MechanicsProf. Y. F. Chen Schrödinger Coherent States of the 1D H.O. Quantum Harmonic Oscillator

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

20 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 MechanicsProf. Y. F. Chen Schrödinger Coherent States of the 1D H.O. Quantum Harmonic Oscillator

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

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

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

24 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 MechanicsProf. Y. F. Chen Creation & Annihilation Operators Quantum Harmonic Oscillator

25 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 MechanicsProf. Y. F. Chen Creation & Annihilation Operators Quantum Harmonic Oscillator

26 is the hermitian conjugate proof 2006 Quantum MechanicsProf. Y. F. Chen Creation & Annihilation Operators Quantum Harmonic Oscillator

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

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

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

30 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 MechanicsProf. Y. F. Chen Creation & Annihilation Operators Quantum Harmonic Oscillator

31 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 MechanicsProf. Y. F. Chen Creation & Annihilation Operators Quantum Harmonic Oscillator

32 is the hermitian conjugate proof 2006 Quantum MechanicsProf. Y. F. Chen Creation & Annihilation Operators Quantum Harmonic Oscillator

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

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

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


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