EP 228: Quantum Mechanics Lecture 11: Classical Vs Quantum - - PowerPoint PPT Presentation

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EP 228: Quantum Mechanics Lecture 11: Classical Vs Quantum Mechanics Probabilistic interpretation For large number of identically prepared system, we can say the probability of measuring enery E n will be |C n | 2 . Recall the definition

SLIDE 1

EP 228: Quantum Mechanics Lecture 11: Classical Vs Quantum Mechanics

SLIDE 2

Probabilistic interpretation

For large number of identically

prepared system, we can say the probability of measuring enery En will be |Cn|2. Recall the definition

Hence the expectation value of

energy will be

SLIDE 3

Classical Mechanics

Dynamical variables A, B in the phase

space ( position x and momentum p of the particles in the system)

Hamiltonian H(x,p) helps in finding

the time evolution of x and p

Poisson bracket Hamiltons eqns

SLIDE 4

Quantum classical analogy

Dynamical variables becomes operators

acting on Hilbert space

Poisson brackets becomes commutator

brackets

We can prove for any two operators
Hence non-zero commutator of two
perators will imply uncertainty principle

SLIDE 5

Commutator properties

Commutator bracket under conjugation

gives a negative sign. To make the commutator hermitean, we introduce i.

For [x,p] commutator, Planck’s constant

h is introduced to make it dimensionless

[AB,CD]=A[B,CD] + [A, CD]B=

AC[B,D]+A[B,C]D+[A,C]DB+C[A,D]B

SLIDE 6

Time evolution of state vectors

Postulate : state vector evolves in

time governed by the time dependent Schrodinger equation

For time independent Hamiltonian, we

can integrate the equation to give

SLIDE 7

Nature of the time evolution operator

Time evolution operator
Takes state vector from to to t
Note that the norm is t independent
Such operators are called unitary opr

SLIDE 8

Free Particle

Let us work in position basis
Since the Hamiltonian is dependent
n p, we can work in momentum basis

SLIDE 9

Free particle continued

The momentum basis

SLIDE 10

Position dependent V(x)

If we operate Hamiltonian on position

basis

We will show it will be differential
perator at xo

SLIDE 11

Momentum operator

Expand commutator in
Simplifies to
Recall derivative of delta function

SLIDE 12

Momentum operator contd

Hence we can write

SLIDE 13

Position opr acting on p basis?

Do a similar exercise and show that
With this data,
Show that the above equation is time