time dependence of the statistical distribution, constants of the motion

Consider the Schrödinger equation and the complex conjugate equation:

∂Ψ-- ∂Ψ-∗ ∗ iℏ ∂t = H Ψ, iℏ ∂t = − (H Ψ )

If Ψ is normalized to unity at the initial instant, it remains normalized at any later time. The mean value of a given observable A is equal at every instant to the scalar product

∫ ∗ < A >= < Ψ, AΨ >= Ψ A Ψd τ

and one has

d ⟨ ∂Ψ ⟩ ⟨ ∂ Ψ ⟩ ⟨ ∂A ⟩ -- < A >= ---,A Ψ + Ψ,A ---- + Ψ, ---Ψ dt ∂t ∂t ∂t

The last term of the right-hand side, < ∂A∕∂t >, is zero if A does not depend upon the time explicitly.

Taking into account the Schrödinger equation and the hermiticity of the hamiltonian, one has

⟨ ⟩ d- 1- 1- ∂A- dt < A >= − iℏ < H Ψ, AΨ > + iℏ < Ψ, AH Ψ > + ∂t

d 1 ⟨ ∂A ⟩ -- < A >= -- < Ψ, [A, H ]Ψ > + --- dt iℏ ∂t

Hence we obtain the general equation giving the time-dependence of the mean value of A:

⟨ ⟩ d- ∂A- iℏdt < A >= < [A, H ] > +i ℏ ∂t

(1)

When we replace Aby the operator e^iξA, we obtain an analogous equation for the time-dependence of the characterisic function of the statistical distribution of A.

In particular, for any variable C which commutes with the Hamiltonian

[C, H ] = 0

and which does not depend explicitly upon the time, one has the result

-d < C >= 0 dt

The mean value of C remains constant in time. More generally, if C commutes with H, the function e^iξC also commues with H, and, consequently

-d iξC dt < e >= 0

The characteristic function, and hence the statistical distribution of the observable C, remain constant in time.

By analogy with Classical Analytical mechanics, C is called a constant of the motion. In particular, if at the initial instant the wave function is an eigenfunction of C corresponding to a give eigenvalue c, this property continues to hold in the course of time. One says that c is a ”good quantum number”. If, in particular, H does not explicitly depend upon the time, and if the dynamical state of the system is represented at time t₀ by an eigenfunction common to H and C, the wave function remains unchanged in the course of time, to within a phase factor. The energy and the variable C remain well defined and constant in time.

1.1 References

[1] Messiah, Albert. ”Quantum mechanics: volume I.” Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.

This entry is a derivative of the Public domain work [1].

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