2-2

Moving Expectation Values

Nicholas Wheeler
Reed College Physics Department
4 February 2010

We are concerned with a quantum system the state vectors of which live in a 3D complex vector space, in which we have erected an orthonormal basis which serves to cast all abstract statements into explicit matrix-theoretic form. We have adopted units which serve to render ℏ = 1.

We bring the system into interaction with an A-meter the matrix representation of which is

𝔸=(({{3, 0, 0}, {0, 3, 2}, {0, 2, 1}}));

and the output dial of which is calebrated

Eigenvalues[𝔸]
N[%]

At time t = 0 the meter reads "3",  thus announcing that the state prepared by the meter is

(Subscript[ψ, 0])=(({{1}, {0}, {0}}));

The motion of the state vector, subseqent to the measurement and up until the the time of the next measurement, is generated by the Hamiltonian H represented by the hermitian matrix

ℍ=(({{1., 2., 3.}, {2., 4., 5.}, {3., 5., 6.}}));

State motion is dictated by the t-dependent unitary matrix

𝕌[t_]:=MatrixExp[-I ℍ t]

𝕌[t]//Chop

ψ[t_]:=𝕌[t].(Subscript[ψ, 0])

The standard Mathematica command Conjugate gets confused re the reality of t, so we introduce

CompConj[A_] := A /.Complex[0,n_]->-Complex[0,n]

MovingExpectedValue[t_]:=(Chop[CompConj[Transpose[ψ[t]]].𝔸.ψ[t]]//Chop)[[1]][[1]]

ExpectedAValue[t_]:=MovingExpectedValue[t]//Simplify//Chop

ExpectedAValue[t]

Plot[Evaluate[ExpectedAValue[t]],{t,0,20}]

The eigenfrequencies (since we have set ℏ = 1) are

{(Subscript[ω, 1]),(Subscript[ω, 2]),(Subscript[ω, 3])}=Abs[Eigenvalues[ℍ]]

So the interference frequencies are

{(Subscript[Ω, 1]),(Subscript[Ω, 2]),(Subscript[Ω, 3])}={(Subscript[ω, 1])-(Subscript[ω, 2]),(Subscript[ω, 1])-(Subscript[ω, 3]),(Subscript[ω, 2])-(Subscript[ω, 3])}

and the associated periods are

{(Subscript[τ, 1]),(Subscript[τ, 2]),(Subscript[τ, 3])}=((2π)/({(Subscript[Ω, 1]),(Subscript[Ω, 2]),(Subscript[Ω, 3])}))

which are evident in the figure. 

A time-average of the expected value is given by

((Chop[(∫_0^20)ExpectedAValue[t]\[DifferentialD]t])/20)

Here I redraw the figure with gridlines indicating the principal period ((Subscript[τ, 3])) and preceding time-averaged expected value:

MovingMean=Plot[Evaluate[ExpectedAValue[t]],{t,0,20}, GridLines->{{18.22},{1.94}}]

At the moment, I cannot account for the fact that (Subscript[τ, 3]) appears to be twice the period of the oscillation most evident in the figure.

Here I do some informal "eyeball spectral analysis" of the figure:

SmoothedApprox=Plot[1.94+1.2Cos[4π(t/18.22)],{t,0,20}, GridLines->{{18.22},{1.94}}]

Show[MovingMean,SmoothedApprox]

Effect of altering the initial state:

Suppose we alter the initial state, from (({{1}, {0}, {0}})) to

(Subscript[ψ, 0])=(({{0}, {1}, {0}}));

Proceeding exactly as before…

ψ[t_]:=𝕌[t].(Subscript[ψ, 0])

MovingExpectedValue[t_]:=(Chop[CompConj[Transpose[ψ[t]]].𝔸.ψ[t]]//Chop)[[1]][[1]]

ExpectedAValue[t_]:=MovingExpectedValue[t]//Simplify//Chop

ExpectedAValue[t]

we are led to this variant of the preceding figure:

Plot[Evaluate[ExpectedAValue[t]],{t,0,20}]

The time-averaged mean assumes a value

((Chop[(∫_0^20)ExpectedAValue[t]\[DifferentialD]t])/20)

which differs from its former value (1.94261)

Plot[Evaluate[ExpectedAValue[t]],{t,0,20},GridLines->{{18.22},{2.38}}]

Effect of altering the dimension of the state space:

If the state vector lives in a n-dimensional vector space, then ℍ is n×n. In the absence of spectral degeneracy, the system posseslses n eigenfrequencies

{(Subscript[ω, 1]),(Subscript[ω, 2]),...,(Subscript[ω, n])}

from which one can construct interference frequencies

(Subscript[Ω, ij])=(Subscript[ω, i])-(Subscript[ω, j])

that are

f[n_]:=(1/2)n(n-1)

in number—a number that increases rapidly, and that is equal to n only in the case n = 3:

Table[{n,f[n]},{n,1,10}]//TableForm

We therefore expect the motion of the expectation values to become rapidly more complicated as n is increased. Which I illustrate by example:

𝔸=(({{3, 0, 0, 0}, {0, 3, 2, 1}, {0, 2, 3, 2}, {0, 1, 2, 2}}));
(Subscript[ψ, 0])=(({{1}, {0}, {0}, {0}}));
ℍ=(({{1., 2., 3., 4.}, {2., 5., 6., 7.}, {3., 6., 8., 9.}, {4., 7., 9., 0.}}));
𝕌[t_]:=MatrixExp[-I ℍ t]

ψ[t_]:=𝕌[t].(Subscript[ψ, 0])

MovingExpectedValue[t_]:=(Chop[CompConj[Transpose[ψ[t]]].𝔸.ψ[t]]//Chop)[[1]][[1]]

ExpectedAValue[t_]:=MovingExpectedValue[t]//Simplify//Chop

ExpectedAValue[t]

Plot[Evaluate[ExpectedAValue[t]],{t,0,25}, PlotRange->{0,4.0}]

{(Subscript[ω, 1]),(Subscript[ω, 2]),(Subscript[ω, 3]),(Subscript[ω, 4])}=Abs[Eigenvalues[ℍ]]

{(Subscript[Ω, 1]),(Subscript[Ω, 2]),(Subscript[Ω, 3]),(Subscript[Ω, 4]),(Subscript[Ω, 5]),(Subscript[Ω, 6])}={(Subscript[ω, 1])-(Subscript[ω, 2]),(Subscript[ω, 1])-(Subscript[ω, 3]),(Subscript[ω, 1])-(Subscript[ω, 4]),(Subscript[ω, 2])-(Subscript[ω, 3]),(Subscript[ω, 2])-(Subscript[ω, 4]),(Subscript[ω, 3])-(Subscript[ω, 4])}

The curve is—as anticipated—now much more complicated. 

I cannot presently explain why the period most evident in the figure (τ ≈ 11) does not appear in the list of interference periods:

{(Subscript[τ, 1]),(Subscript[τ, 2]),(Subscript[τ, 3]),(Subscript[τ, 4]),(Subscript[τ, 5]),(Subscript[τ, 6])}=((2π)/({(Subscript[Ω, 1]),(Subscript[Ω, 2]),(Subscript[Ω, 3]),(Subscript[Ω, 4]),(Subscript[Ω, 5]),(Subscript[Ω, 6])}))