From Classical Mechanics to Quantum Field Theory

A Short Course on Quantum Mechanics and Methods of Quantization 9

the classical configuration space of the system:

H=

{

ψ(x):

∫

D

dx|ψ(x)|^2 <∞

}

≡L^2 (D). (1.7)

In this case,ρ(x)≡|ψ(x)|^2 is interpreted as a probability density,|ψ(x)|^2 dx
representing the probability of finding our system in the volume [x, x+dx]ofthe
classical configuration space. The functionψ(x) is called a wave-function.

1.2.1.2 Observables

Postulate 2. The space of observablesOof a QM system is given by the set of
all self-adjoint operators onH.
This means that every observableOˆhas a real spectrum and admits a spectral
decomposition^6 :

Oˆ=

∑

λ

λPˆλ, Pˆλ≡|ψλ〉〈ψλ|, (1.8)

where|ψλ〉is the eigenvectors ofOˆcorresponding to the eigenvalueλ:Oˆ|ψλ〉=
λ|ψλ〉.Since{|ψλ〉}λform an orthonormal basis forH,wehavetherelations:

PˆλPˆμ=δλμPˆλ, (1.9)

∑

λ

Pˆλ=I. (1.10)

Given a pure state|ψ〉∈H, one can consider the mean value (ore expectation
value) of an observableOˆover a state defined as:
〈
Oˆ

〉

≡〈ψ|Oˆ|ψ〉=Tr

[

ρψOˆ

]

. (1.11)

Notice that (1.11) defines a paring betweenHandO(quadratic in the vectors
and linear in the observables), which depends on the scalar product, which we will
discuss in Subsect. 2.1.4.
One can also consider the variance ofOˆover the state|ψ〉as given by:

Oˆ≡

〈

Oˆ^2

〉

−

〈

Oˆ

〉 2

. (1.12)

(^6) Strictly speaking, in writing expression (1.8) we have assumed the spectrum to be discrete.
An analogous formula holds in the case of a continuous spectrum with the sum replaced by the
integral over the spectral measure. For simplicity, we will stick to the discrete notation also in
the rest of this part.