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: