Systems with N Degreesof Freedom
So far, we have restricted our attention (apart from minor digressions) to a system with one degree of freedom, namely, a single particle in one dimension. We now consider the quantum mechanics of systems with N degrees of freedom. The increase in degrees
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The Two-Particle Hilbert Space Consider two particles described classically by (xi, pi) and (x2, p 2). The rule for quantizing this system [Postulate II, Eq. (7.4.39)] is to promote these variables to quantum operators (XI, P 1) and (X2 , P 2 ) obeying the canonical commutation relations: (i= I, 2)
(IO.l.la) (IO.l.lb) (IO.l.lc)
It might be occasionally possible (as it was in the case of the oscillator) to extract
all the physics given just the canonical commutators. In practice one works in a basis, usually the coordinate basis. This basis consists of the kets lxix2) which are
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simultaneous eigenkets of the commuting operators X 1 and X 2 :
CHAPTER 10
Xdx1x2) =xdx1x2)
X2l x1x2) = x2lx1x2)
(10.1.2)
and are normalized ast (10.1.3)
In this basis
(10.1.4)
We may interpret (10.1.5)
as the absolute probability density for catching particle 1 near x 1 and particle 2 near x 2 , provided we normalize IIJI) to unity (10.1.6)
There are other bases possible besides lx 1x 2). There is, for example, the momentum basis, consisting of the simultaneous eigenkets IP1P2) of P1 and P2. More generally, we can use the simultaneous eigenkets lm1m2) of two commuting operators§ QI(XI' PI) and Q2(X2, P2) to define then basis. We denote by 'W'I®2 the two-particle Hilbert space spanned by any of these bases. 'W' 102 As a Direct Product Space
There is another way to arrive at the space 'W' 102 , and that is to build it out of two one-particle spaces. Consider a system of two particles described classically by (x 1 , pi) and (x 2 , p 2). If we want the quantum theory of just particle 1, we define operators X1 and P1 obeying [XI, PI]= iii!
(10.1.7)
The eigenvectors lx 1 ) of X 1 form a complete (coordinate) basis for the Hilbert space j: Note that we denote the bra corresponding to lxlxD as (xix21. §Note that any function of X, and P, commutes with any function of X 2 and P 2 •
\/ 1 of particle l. Other bases, such as lp 1) of P 1 or in general, lm 1) of 0 1(X 1, P1) are also possible. Since the operators X 1 , P 1 , 0 1 , etc., act on \/ 1 , let us append a superscript (1) to all of them. Thus Eq. ( 10.1.7) reads
(10.1.8a) where /( 1l is the identity operator on \1 1. A similar picture holds for particle 2, and in particular, ( 10.1.8b) Let us now turn our attention to the two-particle system. What will be the coordinate basis for this system? Previously we assigned to every possible outcome 1 of a position measurement a vector lx 1 ) in \/ 1 and likewise for particle 2. Now a position measurement will yield a pair of numbers (x1, x2). Since after the measurement particle 1 will be in state Ix 1 ) and particle 2 in Ix 2 ), let us denote the corresponding ket by lxi)®Ixz):
x
IX1 >Q9 IX2 >.,_. {particle 1 at x 1 particle 2 at
Xz
( 10.1.9)
Note that lx1)®lx2) is a new object, quite unlike the inner product (lf/d lf/2) or the outer product llf/ 1 )(1f/ 2 1 both of which involve two vectors from the same space. The product lx1)®lx2 ), called the direct product, is the product of vectors from two different spaces. T
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