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Commutators
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Define the commutator as: [A,B] = AB - BA
In quantum mechanics, two quantities that can be simultaneously determined
precisely have operators which commute. Consider position, x, and momentum
p -> -ih∂/∂x
When dealing with differential operators, we need a dummy function, f, on
which to operate.
[x,p]f = (xp - px)f
= x(-ih∂f/∂x) - (-ih(∂f/∂x)x
= -ihx∂f/∂x + ih∂(xf)/∂x
= -ihx∂f/∂x + ihx(∂f/∂x) + ihf(∂x/∂x)
= ihf
Therefore,
[x,p] = ih
What this means is that you cannot simultaneously measure distance and
momentum with precision. This is the essence of the Heisenberg
Uncertainty principle.
Angular momentum, L:
Classically,
L = R x P where L points along axis of rotation.
The cross product in cartesian coords is:
L_{x} = yp_{z} - zp_{y}
L_{y} = zp_{x} - xp_{z}
L_{z} = xp_{y} - yp_{x}
[L_{x},L_{y}] = [yp_{z} - zp_{y}, zp_{x} - xp_{z}]
= -ihyp_{x} + ihxp_{y}
= ih(xp_{y} - yp_{x})
= ihL_{z}
[L_{y},L_{z}] = ihL_{x}
[L_{z},L_{x}] = ihL_{y}
What this means is that you cannot simultaneously measure angular
momentum in more than one direction
Spin Angular momentum, S:
- - - - - - - -
[σ_{x},σ_{y}] = | 0 1 || 0 -i | - | 0 -i || 0 1 |
_{ }| 1 0 || i 0 | | i 0 || 1 0 |
- - - - - - - -
- - - -
_{ }= | i 0 | - | -i 0 |
_{ } | 0 -i | | 0 i |
- - - -
- -
_{ } = 2ih | 1 0 |
_{ } | 0 -1 |
- -
_{ }= 2ihσ_{z}
Or we can write,
[S_{x},S_{y}] = ihS_{z} since S = hσ/2
In general we can sumarize the commutation relations as follows:
[x_{i},x_{j}] = 0
[p_{i},p_{j}] = 0
[x,p_{x}] = ih
Or in Kronecker Delta form:
[x_{i},p_{j}] = ihδ_{ij}