# ps6 - E x ( z ) is ˆ E x ( z ) = E sin( kz ) ± a + a †...

This preview shows pages 1–2. Sign up to view the full content.

Harvard University Physics 143b: Quantum Mechanics II Problem Set 6 due Friday November 5 Problem numbers refer to Griﬃths, second edition 1. Griﬃths 9.11 2. Griﬃths 9.14 (read Section 9.3.2 and 9.3.3, including material not covered in class). 3. Griﬃths 9.21 Do this by the quantized electric ﬁeld method developed in class. We wrote the electric ﬁeld as ~ E ( ~ r ) i X ~ k,s ~e ~ k,s ± a ~ k,s e i ~ k · ~ r - a ~ k,s e - i ~ k · ~ r ² where ~e ~ k,s is the polarization vector, and a ~ k,s is the photon annihilation operator. We approximated e i ~ k · ~ r 1 in class. Here keep the next term with e i ~ k · ~ r 1 + i ~ k · ~ r. Hence obtain the answer in (9.95) with ˆ n the polarization vector. Proceed using (9.95) to work out (b) and (c). 4. Consider an optical cavity with a single mode of the electromagnetic ﬁeld, with an electric ﬁeld polarized in the x direction, and frequency ω . So the electric ﬁeld operator ˆ

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: E x ( z ) is ˆ E x ( z ) = E sin( kz ) ± a + a † ² where E is a normalization constant, and a is the photon annihilation operator. We also deﬁne the photon number operator ˆ n by ˆ n = a † a. (a) Let the state of cavity ﬁeld at time t = 0 be | Ψ(0) i = | m i , where m is an integer. Determine | Ψ( t ) i . Then compute the average values of ˆ E x and ˆ n at time t , along with their variances σ E x ( t ) and σ n ( t ) as deﬁned in Section 3.5.1. (b) As above, but for | Ψ(0) i = ( | m i + e iϕ | m + 1 i ) / √ 2, where m is a positive integer, and ϕ is a phase. 1 (c) Use (3.62) to establish an uncertainty relation between the electric ﬁeld and the photon number. Verify that your answers above are consistent with this uncer-tainty relation. 2...
View Full Document

## This note was uploaded on 02/15/2011 for the course PHYS 143B taught by Professor Unknow during the Spring '10 term at Harvard.

### Page1 / 2

ps6 - E x ( z ) is ˆ E x ( z ) = E sin( kz ) ± a + a †...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online