Harmonic Oscillator

Question 1

Harmonic oscillator in classical Hamiltonian

Answer 1

Here U(x) is such that it’s min is at x=0 (differentiate 2nd term)

Question 2

Hamilton equations for classical harmonic oscillator

Answer 2

Question 3

General solution of classical harmonic motion (x)

Answer 3

Where differential equation is derived from Hamiltons equations

Question 4

General solution of classical harmonic motion (p)

Answer 4

Question 5

Quantum mechanical harmonic oscillator (without using N or a)

Answer 5

Question 6

Answer 6

Question 7

Answer 7

Question 8

Answer 8

Question 9

Answer 9

Both ways around, this is a Hermitian operator

Question 10

Write Hamiltonian operator for harmonic oscillator in terms of a^^

Answer 10

Question 11

Prove that all eigenvalues of N^^ are non negative

Answer 11

Determine spectrum and eigenstates of N first

Question 12

Answer 12

Question 13

What does this mean

Answer 13

|Ψ> is an eigenstate of N^^ with eigenvalue λ

Question 14

Answer 14

This means that either a^^|Ψ> =0 or a^^|Ψ> is an eigenstate of N^^ with eigen value λ-1

Question 15

What are possible values of λ

Answer 15

In second case we will eventually obtain neg values which is not possible as all eigenvalues of N^^ are non neg => stronger version of this lemma

Question 16

Stronger version

Answer 16

Question 17

Answer 17

Question 18

Answer 18

Question 19

Answer 19

(Remember, this only pertains to harmonic oscillator)

Question 20

Answer 20

The ground state

Question 21

Ground state

Answer 21

Also called vacuum state

Question 22

From ground state, construct states

Answer 22

Question 23

|n> is and eigenstate of N^^ with eigenvalue n, in Dirac notation

Answer 23

Question 24

Finish solution to spectral problem for N^^

Answer 24

Question 25

Every eigen state of N^^ is?
Prove?

Answer 25

An eigenstate of H^^

Question 26

The eigenstates of H^^ for Harmonic Oscillator are?
And the corresponding eigenvalues are?

Answer 26

Question 27

Using definition of a^

Answer 27

Question 28

Rewrite this according to the definitions of p^^ and x^^

Answer 28

Question 29

Solution to

Answer 29

Question 30

Normalise

Answer 30

Question 31

Answer 31

Question 32

|n> =? (Value)

Answer 32

Question 33

Answer 33

Question 34

Determine proportionality coefficient of below

Answer 34

Question 35

|1> =? |2> =? |n> =?

Answer 35

Question 36

If |Ψ> is an eigenstate of N then?

Answer 36

Either:

λ = 0 and a|Ψ > = 0

Or

λ != 0 , a|Ψ> != 0 and
a|Ψ> is an eigenstate of N with eigenvalue λ - 1

Question 37

Then a^dagger|Ψ> ?

Answer 37

!= 0

And a^dagger|Ψ> is an eigenstate of N with eigen value λ + 1

Question 38

Hamiltonian of harmonic oscillator

Answer 38

Question 39

Important trick for solving equations with a and a^dagger

Answer 39

Use commutator with trivial separation of factors

Question 40

Answer 40

Question 41

Answer 41

Question 42

Answer 42