QuantumPhysics.dvi

so that|φ(k)|= 1. The phase ofφ(k) is not determined by the normalization conditions

(5.58). The simplest choice is given byφ(k) = 1, and it is this choice that we adopt.

• Matrix elements of the operatorsXandPbetween a general state|ψ〉and the position

or momentum eigenstates may be derived in a similar manner. It is customary to define the

position and momentum spacewave functionsrespectively by,

ψ(x) ≡ 〈x;X|ψ〉

ψ ̃(k) ≡ 〈k;P|ψ〉 (5.65)

All other matrix elements may be expressed in terms of these wave functions. Thus, one has

for example,

〈x;X|X|ψ〉 = xψ(x)

〈k;P|P|ψ〉 = ̄hkψ ̃(k)

〈x;X|P|ψ〉 = −i ̄h

∂

∂x

ψ(x)

〈k;P|X|ψ〉 = i

∂

∂k

ψ ̃(k) (5.66)

Matrix elements between general states|φ〉and|ψ〉, with wave functions respectivelyφ(x),

andψ(x), may be calculated analogously,

〈ψ|X|φ〉 =

∫

R

dx ψ(x)∗〈x;X|X|φ〉=

∫

R

dx xψ(x)∗φ(x)

〈ψ|P|φ〉 =

∫

R

dx ψ(x)∗〈x;X|P|φ〉=−i ̄h

∫

R

dx ψ(x)∗

∂φ(x)

∂x

(5.67)

5.6 The harmonic oscillator

The 1-dimensional harmonic oscillator for the positionXand momentum P operators is

given by the Hamiltonian

H=

1

2 M

P^2 +

1

2

Mω^2 X^2 (5.68)

where [X,P] =i ̄h. The constantsM andω are respectively the mass and the frequency,

while the combinationK=Mω^2 is the spring constant. The importance of the harmonic

oscillator derives from the fact that it may be used as an approximation for the Hamiltonian

with a general potentialV, considered around one of the minimax 0 ofV, where,

V(x) =V(x 0 ) +

1

2

V′′(x 0 )(x−x 0 )^2 +··· (5.69)