Introduction to Commutators

The general criterion for simultaneous measurement: 

First we make one measurement which must determine the exact meaning of measured quantity. In mathematical terms, the wavefunction y must be an eigenfunction in the following form

Ay  =  Ay
 

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

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

Ay = Ay  and   By = By

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

BAy- ABy  =  ABy- BAy  =  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:

Momenum (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:

Momentum (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 don't commute with 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 know the operators x, t, p, H. We obtain the kinetic moment operator L using the 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*AydV) <A> should be real, i.e. <A> = <A>*

ò y*AydV  =  ò y*A*y*dV     or     A  = A†

in general:

ò y*AφdV  =  ò fA*y*dV

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

ò y*AφdV  =  ò fA*y*dV
 
 

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