I know that it is impossible, according to the laws of quantum mechanics, to know everything about one particle.

Because of this, there are certain combinations of things we cannot know about a single particle.

The problem is, I forget which these are.

There is more than one pair of things we cannot know, right?

What are some of the pairs of things we cannot know?

Also, for any of these pairs of things we cannot know, how could I build/buy a machine to measure each?

You are talking about

*conjugate *variables. Examples:

p

_{α } and r

_{α}where p

_{α } means the generic component α of the vector momentum

**p**, and r

_{α} means the generic component α of the position vector

**r**; so: p

_{x} and x, p

_{y} and y, p

_{z} and z;

more generally, p

_{i} and q

_{i} where q

_{i} is the i-esim generalized lagrangian coordinate and p

_{i} its correspondin generalized momentum, defined as p

_{i} = ∂L/∂q'

_{i}, where L = L(q

_{1}, q

_{2},...q

_{n}, q'

_{1}, q'

_{2},...q'

_{n}, t) is the lagrangian and q'

_{i} stays for time derivative of q

_{i}. p

_{i} and q

_{i} forms a couple of conjugate variables in the

*phase space*, which represent an example of "symplectic coordinates".

Another example of conjugate variables are the angle θ of rotation around an axis and the angular momentum with respect to that axis.

I didn't write energy and time as conjugate variables because time it's not an operator in quantum mechanics.

The uncertainty (or indeterminacy) principle in its exact form, infact, refers to operators, more precisely self-adjoint operators in a Hilbert space, which corresponds to the dynamical variables, that is to the physical quantities.

The principle actually becomes just a theorem in the formal description of quantum mechanics which is based on the Postulates (that is, the most accepted present formulation of QM).

Then the "principle" which becomes a theorem just states that there are couples of such operators which doesn't

*commute *with each other, that is, if A and B are such operators, then A•B ≠ B•A, where • means

*composition * of operators; that is , if you apply first the operator B to a quantum state, described by a vector |ψ> in the Hilbert space (ψ is the "wavefunction"), and then you apply the operator A to the resulting state, that's different than the state you get applying first the operator A and then the operator B:

A•B|ψ> ≠ B•A|ψ>

example: let A be the operator x, that is the operator which multiplies the quantum state (also called "ket") |ψ> by the coordinate x:

x|ψ> = x*|ψ>

if you remember something about vectorial spaces, you can multiply a scalar by a vector getting another vector with the same direction of the first but with a different modulus, which is the modulus of x times the modulus of the previous vector; that's what it means (at least if x is a real number; if it's complex, then the resulting vector is also rotated of an angle corresponding to the argument of the complex number).

let instead B be an operator of a different kind, the operator d/dx, that is, the operator which transforms the vector |ψ> in its derivative with respect to x (in general, the wavefunction ψ depends on x, y, z and time t):

(d/dx)|ψ> = d|ψ>/dx

Now let's see what happens applying in sequence the two operators:

1. x•(d/dx)|ψ> = x d|ψ>/dx = x*d|ψ>/dx

2. (d/dx)•x|ψ> = (d/dx) x*|ψ> = (rule of derivative of a product of functions) =

= (dx/dx)*|ψ> + x*d|ψ>/dx = 1 + x*d|ψ>/dx

which is

*different* from x*d|ψ>/dx.

So A•B|ψ> ≠ B•A|ψ> which can also be written: A•B|ψ> - B•A|ψ> ≠ 0.

So, the two operators don't commute, which is also written in the more coincise form:

[A,B] ≠ 0 where [A,B] means A•B - B•A

From the fact that two operators don't commute with each other it can be demonstrated that the

*statistical * indeterminacy ΔA = standard deviation of the physical quantity associated to the operator A, multiplied by the

*statistical * indeterminacy ΔB = standard deviation of the physical quantity associated to the operator B, cannot be less than a specific amount:

ΔA*ΔB ≥

~~h~~/2

where

~~h~~ = h/2π and h is Plank's constant.

In QM you will discover that the operator d/dx is associated to the physical quantity momentum along x, that is p

_{x}, so the indeterminacy principle about x and p

_{x} (or about another couple of conjugate operators) has become a theorem between operators in the Hilbert space! Magic of mathematics!