Lecture 15

Lecture Lecture 14: Time-dependent Time-dependent perturbation perturbation theory: correspondence principle + sudden approximation (11/10/2005) Note about the last time: I was cheating a bit with the heuristic derivation of the Casimir effect. The Casimir energy in the three-dimensional case is actually proportional to n3 = ζ ( 3) = 1/120, not to ζ  to  ζ (( 1). The particular constant ζ  constant  ζ (( 1) is relevant in the one-dimensional case which can still be experimentally measured.

−

−

−

− −

−

The correspondence principle requires that if all the quantum numbers are large, the conclusions of quantum mechanics reduce to classical (non-quantum) physics. We want to check this statement for the Larmor formula, a classical formula for power emitted by an accelerating charge.

Additiona Additionall reading reading if you you wish: Bran Bransden sden & Joacha Joachain in ch. 9.5 Later, we will talk about the sudden approximation, a way to calculate how the system responds if the Hamiltonian Hamiltonian  H 0  in the past is suddenly changed to H  to  H 0  in the future. 

Larmor and the quantum harmonic oscillator According to classical physics, a charge q  charge  q  accelerating  accelerating by  a  emits the power (energy per unit time) q 2 a2 P  = 6π0 c3 For large quantum numbers in ordinary systems, quantum physics should normally reduce to classical physics physics because of the so-called so-called corresponde correspondence nce principle. principle. Is it true for the Larmor formula formula above? We will check it in the case of the dipole radiation of the harmonic oscillator – a charged microobject attached to a “spring”. Recall that the dipole matrix element is   =  q n |xˆ|n, P  

ˆ  = x

which means that

1 √  (ˆa ω 2m

a− ) +  + ˆ

√  √ 

√ 

¯ω q  q  hω h   = (δ n ,n+1 +  δ n ,n n (ˆa+  + ˆa ) n = ,n+1 n + 1 + δ  ω 2m ω 2m

P 

√   | 

−

|





1

−

√ n)

The two terms are responsible for absorption ( n  of the harmonic oscillator goes up) or emission (n (n goes down), respectively. You see that the matrix element of the dipole is only nonzero if  n if  n = n 1. It means that we can only emit or absorb a single frequency in this dipole radiation approximation 

¯ ωrad  = E   =  E n hω h



±



− E   = hω( ¯hω(n + 1/ 1/2) − hω( ¯ ω(n + 1/ 1 /2) = ±hω, ¯ ω, h h n

namely the frequency of the harmonic oscillator itself. We have calculated the rate  A of  A  of the spontaneous emission in two different ways – lectures 12 and 14. The general result can be specialized to our case of the harmonic oscillator with n with  n =  = n  n 1:

−

Rn,n

ω3 2 ω3  = = 1 3π0hc ¯hc3 3π0hc ¯ c3 h

P 

−

2

 q  hnh¯n  2mω

=

q 2 nω 2 . 6π0 mc3

This is the probability per unit time, and therefore the average energy per unit time is P  =  R n,n

q 2 ω 2 q 2 ω 2 ¯hω  = (nhω) ¯ ω) = (E  h 1 hω = 6π0 mc3 6π0mc3

−

1

− 0.5¯hω) hω)

If we want to compare it with the classical Larmor formula, we also need to know the acceleration a. A harmonic oscillator has x = x 0 cos(ωt) which implies a = ω 2 x. Therefore a 2 = x 20 ω 4 cos2 (ωt). If we average the cos over one period, we finally obtain

−

a  = 12 x ω 2

2 0

4

and because E  = mω 2 x20 /2 – the maximal potential energy when the kinetic energy is zero – it can also be written as Eω 2 2 a = m 2 If we now replace the factors  E ω /m in our quantum mechanical result for P  by a 2 and neglect the 0.5¯hω term (the ground state energy, negligible compared to the total energy in our approximation), we obtain our classical Larmor formula. You can always try these consistency checks that your quantum theory gives you the expected classical results in the appropriate limit.

 

The sudden approximation In our discussion of general time-dependent perturbation theory, we always turned on the perturbation for a finite time interval only. (Except for the harmonic perturbations that are contributing all the time.) But we may try something different now: imagine that the Hamiltonian approaches H 0 for t , but it approaches a different Hamiltonian  H 0 for t + . We want to allow the case in which the difference H 0 H 0  is actually large. Let’s use the notation

→ −∞

→ ∞



−



• |n for the eigenstates of  H  • |n   for the eigenstates of  H  0





0

What’s our task? We want to start, at  t