Parity_Time_Reversal.pdf

7/27/2019 Parity_Time_Reversal.pdf

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Lecture 20

Parity and Time Reversal

November 15, 2009

Lecture 20


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Time Translation

|(t + ) = U[T()] |(t) =I

i

h H

|(t)

H is the generator of time translations.

U[T()] = I ihH

[H, H] = 0 time translational invariance

also [H, H] = 0 H = 0 energy conservat

ihd

dt|(t) = H|(t)

ih

d

dt |(t

) = H|(t

)

but what if someone is at the control panel

changing H as a function of time?

Lecture 20 1


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Time Dependent Hamiltonian

H =p2

2m + V(x, t)

ihd

dt|(t) = H(t)|(t)

ihd

dt |(t) = H(t)|(t)

H(t) = H(t) not time translationally invariant

also ihH = [H, H] +

H

t

= 0

energy is not conserved

Lecture 20 2


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Parity Inversion

x = x p = p

mirror reflection plus rotation by 180o

about axis perpendicular to mirror

|x = | x |p = | p

(x) = x|| = (x)

(p) = p|| = (p)

2 = I eigenvalues of are 1

if +1: (x) = (x) (p) = (p)

if 1: (x) = (x) (p) = (p)

1 = and =

is both unitary and Hermitian

Lecture 20 3


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Parity Conservation

x = x p = p

H(x,p) = H(x, p) H = H

[, H] = 0 [, U(t) ] = 0

parity is conserved

if system is in a state of definite parityit remains in a state of definite parity

If H is non-degenerate and H(x,p) = H(x, p),an energy eigenstate must also be a parity eigenstate.

Lecture 20 4


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Time Reversal

Time reversal is really just reversal of motion.

Under time reversal

x = x p = p

and we also switch initial and final states.

Imagine taking a movie of some process and

then playing the movie backwards.

Does the motion in the backward played movie

obey the laws of physics? If yes, then the

process is invariant under time reversal. If no,then it is non-invariant under time reversal.

Time reversal in quantum mechanics involves

some subtle issues, so, lets study it first in

classical mechanics to help get our bearings.

Lecture 20 5


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Time Reversal in Classical Mechanics

x

(t) = x(t)

x(t) =dx(t)

dt=

dx(t)

d(t)= x(t)

x(t) = dx(t)dt

= dx(t)dt

= dx(t)d(t)

= x(t)

So, under time reversal, the velocity changes sign but

the position and acceleration of the particle do not.

mx(t) = mx(t) = mF(x(t)) = mF(x(t))

x(t) satisfies Newtons Law providedthe force is velocity independent.

Example, of a ball in a gravitational field.

Lecture 20 6


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Particle in a Magnetic Field

A positively charged particle moving in a region

of magnetic field will curve in a counter-clockwisedirection when viewed in the direction of the magnetic

field. If a film of this is played backwards, the particle

will curve in a clockwise direction in violation of the

physics of electromagnetism and will not be invariant.

What we have done above is time reverse the system,

i.e., the charged particle. We have kept the externally

applied field unchanged. Under this situation, we

have a violation of time reversal invariance. If,

however, we were to consider the entire universe

including the sources of the magnetic field as part of

our system and apply the time reversal transformation

to the whole universe, the motion would be timereversal invariant. The velocities of the charges in

the wire that make up the current that produces

the magnetic field in the time reversed movie would

change direction and the current and, therefore, the

magnetic field would be reversed.

The laws of physics are almost all completely invariant

when time reversal is applied to the whole universe

but there is a small violation due to the weak

interaction!

Lecture 20 7


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Anti-Unitary Operator

must be an anti-unitary operator.

It must contain the complex conjugation operator.

= UK

where U is a unitary operator and K is the complex

conjugation operator.

then iH = iH = iH

H = H and H commute

This tells us that the Hamlltonian must be real

in order for time reversal invariance to hold.

Lecture 20 9


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The Complex Conjugation Operator

| = | = UK|

=i

i|U|i =i

|i U|i

|

= | =U

K|

=i

i|U|i =i

|i U|i

| = ij

|ij|UU|ij|

=i

j

|ij|ij| =i

j

|ij|ij

= i

|ii| = |

time reversal exchanges initial and final states.

(t)|(t) = (t)|(t)

Lecture 20 10


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Time Reversed Wave Function

|(t) = |(t) =

|xx|(t) dx

= |x(t)|x dx

(x, t) = x||(t) = (t)|x = (x,t)

L 20 11

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