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 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:
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.
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
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
= |
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 = |
E = p²/2m + V | H = − |
Lx = ypz − zpy | Lx = |
Ly = zpx − xpz | Ly = |
Lz = xpy − ypx | Lz = |
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.
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:
Moreover the operation '†' (or hermitian conjunction) means transition from left to right integral:
ò y*AφdV
= ò fA*y*dV