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Chapter 5 1D Harmonic Oscillator.

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Presentation on theme: "Chapter 5 1D Harmonic Oscillator."— Presentation transcript:

1 Chapter 5 1D Harmonic Oscillator

2 Definition Let us consider 1D system with a potential V(x)
This potential can be expanded in the vicinity of a local minimum: Then the QM Hamiltonian of the system will be:

3 5.A.3 Definition This Hamiltonian is time-independent, hence we get a conservative eigenvalue equation: In the representation the Schrödinger equation will be:

4 5.A.3 Definition This Hamiltonian is time-independent, hence we get a conservative eigenvalue equation: In the representation the Schrödinger equation will be: Eigenvalues E are positive (complement MIII) Eigenfunctions φ are of a definite parity – even or odd (complements FII and CV) The spectrum of E is discrete (complement MIII)

5 5.B.1 Eigenvalue problem The Hamiltonian can be parametrized and made dimensionless by introducing operators Then:

6 5.B.1 Eigenvalue problem For the new dimensionless Hamiltonian the eigenvalue problem becomes Let us introduce the following operators Then And

7 5.B.1 Eigenvalue problem Also Therefore It can be shown that

8 5.B.1 Some properties Commutators Also

9 5.B.2 Some properties Since One finds that

10 5.B.2 Some properties Since Similarly

11 Some properties Thereby:
Because of this, operators a† and a are called ladder operators; a† – creation (raising) operator and a – destruction (lowering) operator

12 Some properties Thereby:
What happens if we apply the lowering operator repeatedly? Will we reach energy less than zero? (This would contradict the statement that all the eigenvalues of E are positive) So, does the ground state exist? If it does, what is it’s eigenvalue and eigenvector?

13 5.B.3 Ground state If the ground state exists then the following condition must be satisfied: Does this equation have solutions? In the representation: Solution:

14 5.B.3 Ground state

15 Ground state Using the Schrödinger equation:
5.B.3 Ground state Using the Schrödinger equation: We found the ground state! It is non-degenerate and it is greater than zero!

16 Ground state Using the Schrödinger equation:
5.B.3 Ground state Using the Schrödinger equation: We found the ground state! It is non-degenerate and it is greater than zero!

17 5.B.3 Excited states We can generate the rest of the spectrum

18 5.B.3 5.C.1 Excited states Since the ground state is non-degenerate and the excited states are obtained using ladder operators, the entire spectrum is non-degenerate Thereby, the Hamiltonian by itself constitutes a CSCO

19 5.C.1 Excited states Eigenvectors can be normalized

20 5.C.1 Excited states Eigenvectors can be normalized

21 Properties of eigenstates
5.C.1 Properties of eigenstates The Hamiltonian is Hermitian hence its eigenkets are orthonormal: The Hamiltonian is an observable hence its eigenkets constitute a basis:

22 Properties of eigenstates
5.C.1 Properties of eigenstates Action of observable operators:

23 Properties of eigenstates
5.C.1 Properties of eigenstates Matrix elements:

24 Properties of eigenstates
5.C.1 Properties of eigenstates Matrix elements:

25 Properties of eigenstates
5.C.1 Properties of eigenstates Matrix elements:

26 Eigenfunctions The eigenvectors can be written in the representation
Recall The creation operator in the representation:

27 Eigenfunctions Therefore: The creation operator in the representation:

28 5.C.2 Eigenfunctions Therefore:

29 Eigenfunctions We have explicit expressions for the eigenfunctions:
These expressions contain the Hermite polynomials The polynomials of degree n have the parity of

30 5.C.2 Eigenfunctions Graphically:

31 Properties of X and P operators
Recall Therefore For RMS deviations

32 Properties of X and P operators
Using ladder operators For RMS deviations

33 Properties of X and P operators
Using ladder operators

34 Properties of X and P operators
Using ladder operators

35 Properties of X and P operators
Thereby So

36 Time evolution of mean values
5.D.3 Time evolution of mean values If a certain state of the system at t = 0 is: Then So

37 Time evolution of mean values
5.D.3 Time evolution of mean values Using Ehrenfest’s theorem: Integrating this system of equations, one obtains:


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