Lecture 20. Parity and Time Reversal

Transcription

1 Lecture 20 Parity and Time Reversal November 15, 2009 Lecture 20

2 Time Translation ( ψ(t + ɛ) = Û[T (ɛ)] ψ(t) = I iɛ ) h Ĥ ψ(t) Ĥ is the generator of time translations. Û[T (ɛ)] = I iɛ h Ĥ [Ĥ, Ĥ] = 0 time translational invariance also [Ĥ, Ĥ] = 0 Ĥ = 0 energy conservaton i h d dt ψ(t) = Ĥ ψ(t) i h d dt ψ(t ) = Ĥ ψ(t ) but what if someone is at the control panel changing Ĥ as a function of time? Lecture 20 1

3 Time Dependent Hamiltonian Ĥ = ˆp2 2m + V (ˆx, t) i h d dt ψ(t) = Ĥ(t) ψ(t) i h d dt ψ(t ) = Ĥ(t ) ψ(t ) Ĥ(t ) Ĥ(t) not time translationally invariant also i h Ĥ = [Ĥ, Ĥ] + Ĥ t 0 energy is not conserved Lecture 20 2

4 Parity Inversion ˆx = ˆx ˆp = ˆp mirror reflection plus rotation by 180 o about axis perpendicular to mirror ˆπ x = x ˆπ p = p ψ (x) = x ˆπ ψ = ψ( x) ψ (p) = p ˆπ ψ = ψ( p) ˆπ 2 = I eigenvalues of ˆπ are ±1 if +1: ψ( x) = ψ(x) ψ( p) = ψ(p) if 1: ψ( x) = ψ(x) ψ( p) = ψ(p) ˆπ 1 = ˆπ and ˆπ = ˆπ ˆπ is both unitary and Hermitian Lecture 20 3

5 Parity Conservation ˆπˆxˆπ = ˆx ˆπ ˆpˆπ = ˆp Ĥ( ˆx, ˆp) = Ĥ(ˆx, ˆp) ˆπĤ ˆπ = Ĥ [ˆπ, Ĥ] = 0 [ˆπ, ˆ U(t)] = 0 parity is conserved if system is in a state of definite parity it remains in a state of definite parity If Ĥ is non-degenerate and Ĥ( ˆx, ˆp) = Ĥ(ˆx, ˆp), an energy eigenstate must also be a parity eigenstate. Lecture 20 4

6 Time Reversal Time reversal is really just reversal of motion. Under time reversal ˆx = ˆx ˆp = ˆp 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, let s study it first in classical mechanics to help get our bearings. Lecture 20 5

7 Time Reversal in Classical Mechanics x (t) = x( t) ẋ (t) = dx( t) dt = dx( t) d( t) = ẋ( t) ẍ (t) = dẋ (t) dt = dẋ( t) dt = dẋ( t) d( t) = ẍ( t) So, under time reversal, the velocity changes sign but the position and acceleration of the particle do not. mẍ (t) = mẍ( t) = mf (x( t)) = mf (x (t)) x (t) satisfies Newton s Law provided the force is velocity independent. Example, of a ball in a gravitational field. Lecture 20 6

8 Particle in a Magnetic Field A positively charged particle moving in a region of magnetic field will curve in a counter-clockwise direction 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 time reversal 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

9 Time Reversal Operator ψ(t) = e iĥt/ h ψ(0) ψ (t) = ˆτ ψ( t) = ˆτe iĥt/ h ψ(0) we will have invariance if ψ (t) = e iĥt/ hˆτ ψ(0) ˆτe iĥt/ h = e iĥt/ hˆτ ˆτiĤ = iĥ ˆτ ˆτ and Ĥ anti-commute This is bad. It leads to negative energies. Consider an energy eigenstate of a free particle. The energy of the time reversed state would be negative. Ĥ ˆτ ψ E = ˆτĤ ψ = Eˆτ ψ Lecture 20 8

10 Anti-Unitary Operator ˆτ must be an anti-unitary operator. It must contain the complex conjugation operator. ˆτ = Û ˆK where Û is a unitary operator and ˆK is the complex conjugation operator. then ˆτiĤ = iˆτĥ = iĥ ˆτ ˆτĤ = Ĥ ˆτ ˆτ and Ĥ commute This tells us that the Hamlltonian must be real in order for time reversal invariance to hold. Lecture 20 9

11 The Complex Conjugation Operator ψ = ˆτ ψ = Û ˆK ψ = i i ψ Û i = i ψ i Û i φ = ˆτ φ = Û ˆK φ = i i φ Û i = i φ i Û i = i φ ψ = i ψ i j i j φ = j i ψ i j Û Û i j φ j ψ i j φ δ ij j = i ψ i i φ = ψ φ time reversal exchanges initial and final states. φ (t) ψ (t) = ψ( t) φ( t) Lecture 20 10

12 Time Reversed Wave Function ψ (t) = ˆτ ψ( t) = ˆτ x x ψ( t) dx = x ψ( t) x dx ψ (x, t) = x ˆτ ψ( t) = ψ( t) x = ψ (x, t) Lecture 20 11