Introduction to Commutators

The general criterion for simultaneous measurement:

First we make one measurement which must determine the exact meaning of measured quantity, i.e. in mathematical way: wavefunction y requires to have eigenfunction form the following expression

A y = Ay

(A is the operator of measured quantity A, for instance A = h/i∂/∂x for A = p)

Now we make second measurement which must determine the exact meaning of the next measured quantity. Now we demand for the wavefunction y:

A y = Ay and B y = By

Acting by B and A on first and second equations correspondingly one can obtain the following difference:

BA y- AB y = AB y- BA y = AB − BA

(AB − BA)y = 0 · y

In order to measure both quantities A and B simultaneously it should be true for operators A and B:

AB −BA = 0

The quantity AB −BA is referred to as commutator and is also assigned using square brackets as [A, B] = AB− BA.

1. Example:

Impulse (p = h/id/dx) and kinetic energy (Ekin = −h/2md²/dx²) can be accurately measured simultaneously

(p . Ekin- Ekin·p)y = h/i(d/dx)·(−h/2md²/dx²)y-(−h/2md²/dx²)·(h/id/dx)y = 0

2. Example:

Impulse (p) and coordinate (x);

(px −xp)yº (h/i.d/dx x − x h/id/dx)y

= h/i(d/dx(xy) − x d/dxy)

= h/i(dx/dx·y + x dy/dx− x dy/dx)

= h/iy

[p, x] = px −xp = h/i

Hence impulse and coordinate operator doesn't commutate with each other and they can't be precisely measured simultaneously.

That's why we need only commutator [A, B] with operators A and B in order one can say about two quantities that they can be accurately measured simultaneously.

We have already known the operators x, t, p, H. We obtain the kinetic moment operator L using classical relation for vector L = r´ p.

Classical | Operator
p = (px, py, pz) | p = h/i(∂/∂x, ∂/∂y, ∂/∂z)
E = p²/2m + V | H = −h²/2mΔ + V
Lx = ypz − zpy | Lx = h/i(y∂/∂z− z∂/∂y)
Ly = zpx − xpz | Ly = h/i(z∂/∂x− x∂/∂z)
Lz = xpy − ypx | Lz = h/i(x∂/∂y− y∂/∂x)

We obtain the following result for simultaneous measurement of different quantities:

[E, t] = i h

[Lx, Ly] = i h Lz

[Ly, Lz] = i h Lx

[Lz, Lx] = i h Ly

The kinetic moment components are not measurable simultaneously but it's possible to measure squared kinetic moment and its component at the same time:

[L2, Lj] = 0 j = x, y, z

We also need "fluctuations" ΔA and ΔB to be negligibly small (almost zero). If the commutator isn't zero then

(DA)²(ΔB)² ³ 1/4<i [A, B]>².

for coordinate x and impulse p it follows [x,p]=ih: Δx²Δp² ³ h²/4

One more note about Operator Features:

Operators should be "linear" operators in order to satisfy the superposition principle (for instance y = φ1 + φ2):

A (φ1 +φ2) = Aφ1 + Aφ2 A (|1> + |2>) = A |1> + A |2> and A (c φ) = cAφ A (c |f>) = cA |f>

These conditions should be true for any functions.

Since the average values are determined by measured meanings (<A> = òy*A ydV) <A> should be real, i.e. <A> = <A>*

ò y*A ydV = ò y*A*y*dV or A = A†

in general:

ò y*AφdV = ò f A*y*dV

Moreover the operation '†' (or hermitian conjunction) means transition from left to right integral:

ò y*AφdV = ò f A*y*dV