The Harmonic Oscillator
In this section I will put the hamwnic oscillator in its place-on a pedestaL Not only is it a system that can be exactly solved (in classical and quantum theory) and a superb pedagogical tool (which will be repeatedly exploited in this text), but it is al
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£= T+
p2 I ." v,=-+-mw x
2m
2
2
(7.1.1)
where w = (k/m) 1/ 2 is the classical frequency of oscillation. Any H::tmiltonian of the above form, quadratic in the coordinate and momentum, will be called the harmonic oscillator Hamiltonian. Now, the mass-spring system is just one among the following family of systems described by the oscillator Hamiltonian. Consider a particle moving in a potential V(x). If the particle is placed at one of its minima x 0 , it will remain there in a state of stable, static equilibrium. (A maximum, which is a point of unstable static equilibrium, will not interest us here.) Consider now the dynamics of this partide as it fluctuates by small amounts near x = x 0 . The potential it experiences may be expanded in a Taylor series:
dvi
d vi
1 2 V(x)= V(xo)+-. (x-xo)+--. (x-x0 ) 2 +· · · dx xo 2! dx 2 , 0
(7.1.2)
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186 CHAPTER 7
Now, the constant piece V(x 0 ) is of no physical consequence and may be dropped. [In other words, we may choose V(x0 ) as the arbitrary reference point for measuring the potential.] The second term in the series also vanishes since x 0 is a minimum of V(x), or equivalently, since at a point of static equilibrium, the force, -dV/dx, vanishes. If we now shift our origin of coordinates to x 0 Eq. (7.1.2) reads (7.1.3)
For small oscillations, we may neglect all but the leading term and arrive at the potential (or Hamiltonian) in Eq. (7.1.1), d 2 V/dx 2 being identified with k=mm 2 • (By definition, xis small if the neglected terms in the Taylor series are small compared to the leading term, which alone is retained. In the case of the mass-spring system, xis small as long as Hooke's law is a good approximation.) As an example of a system described by a collection of independent oscillators, consider the coupled-mass system from Example 1.8.6. (It might help to refresh your memory by going back and reviewing this problem.) The Hamiltonian for this system is
PT
p~
1
2
2
2
2
Yf=-+-+-mm [xi+xz+(xi-xz)] 2m 2m 2 =
Yfi + Yfz + ~mm 2 (XI- xz) 2
(7.1.4)
Now this Yf is not of the promised form, since the oscillators corresponding to Yfi and Yf2 (associated with the coordinates xi and xz) are coupled by the (xi-xz) 2 term. But we already know of an alternate description of this system in which it can be viewed as two decoupled oscillators. The trick is of course the introduction of normal coordinates. We exchange xi and x 2 for (7.1.5a) and (7.1.5b) By differentiating these equations with respect to time, we get similar ones for the velocities, and hence the momenta. In terms of the normal coordinates (and the corresponding momenta), (7.1.6) Thus the problem of the two coupled masses reduces to that of two uncoupled oscillators of frequencies m1 = m = (k/m)I 12 and mu = 3I 12 m = (3k/m)I 12 .
187
Let us rewrite Eq. (7.1.4) as (7.1.7) where Vij are elements of a real symmetric (Hermitian) matrix Vwith the following values: (7 .1.8)
In switching to the normal coordinates x 1 and xu (and p 1 and Pu ), we are going to a basis that diagonalizes V and reduces th
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