The Uncertainty Principle (With Operators)

Friday, November 30, 2018

The Uncertainty Principle (With Operators)

The Heisenberg Uncertainty Principle

We know that momentum and position are incompatible. You can not have perfect knowledge of both. We understood this simply by thinking about how you would go about measuring the position and momentum of a particle. There is also a way to see this mathematically using the complex vector space and operators in quantum mechanics.

For two operators to be compatible they must have common eigenvectors. This means that there are vectors/states for which both observables have a definite values.
If two operators are fully compatible, there are a complete basis of vectors which are eigenvectors of the two operators.

If two operators share eigenvectors (are commutative), then the following is true.
A|n> = k_n|n>
B|n> = k_n|n>

where |n> are the eigenvectors of A and B with eigenvalues k_n

By looking at what happens when we apply both operators to a eigenvector, we can determine the conditions that mean two operators are compatible.

AB|n>
=Ak_B|n>
=k_BA|n>
=k_Bk_A|n>

and so if they have the same eigenvectors, the order in why you apply the operators does not matter.

AB|n> = BA|n>

The necessary and sufficient condition that there is a complete basis of vectors that are eigenvalues of A and B is

AB = BA
AB - BA = 0
[A,B] = 0
The commutator of A and B is 0

So let's check if our operators for position and momentum are compatible.

[X, P]
XP - PX = ?
Xħ-(i∂/∂x) - ħ(-i∂/∂x)X

We also need something to differentiate with respect to x and so let's take a general function f(x)

ħ(X-(i∂/∂x) - (-i∂/∂x)X)f(x)

-iXħ ∂f(x)/∂x + iħ ∂(Xf(x)/∂x
Using the product rule

-iħX∂f(x)/∂x + iħf(x) + iħX∂f(x)/∂x

=iħf(x)

So we know that
(XP - PX)f(x) = iħf(x)

In other words XP - PX does the same thing as multiplying by iħ

[X,P] = iħ

X and P do not commute and this is consistent with what we found about the uncertainty principle previously.

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