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By Cox R.T., Hubbard J.C.

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32 of nine complex dimensions in the system's Hilbert space, with orthogonal vectors δ00, δ01, δ02, δ10 … for (x,y)=(0,0), (0,1), (0,2), (1,0) etc. Each of these nine vectors would possess, simultaneously, exact values for both x and y, and they'd be eigenvectors of both X and Y. ) However, not every state with a definite x would necessarily possess a definite y, or vice versa. For example, (δ10+δ12)/√2 would have a probability of 100% to be found at x=1, but a 50/50 chance of being found either at y=0 or y=2.

We can write the P matrix in terms of its own eigenvectors, to get a much simpler expression for it than the one based on D and Equation (21). Just like Equation (18) for X, we take all the possible values for the momentum, p, and multiply them by tensors that project onto the states with those values of momentum: P = (h√3/4π) (–1 p –1 ⊗p –1 + 0 p 0 ⊗p 0 + 1 p 1 ⊗p 1 ) ( 2 2 ) As with X, this matrix allows us to calculate the average momentum for any state vector ψ, as <ψ,Pψ>. And as with the momentum operator for wave functions, this matrix can be used in matrix equations, similar to the Klein-Gordon and Schrödinger wave equations.

33 = √(<ψ,A2ψ>–<ψ,Aψ>2) ≥ (1/2)|<ψ, [A,B]ψ>| then: ∆a ∆b (24) Unfortunately, P and X in our toy universe actually fail some of the technical requirements needed for this to be true (because x undergoes a sudden jump in value from 3 back to 0, which complicates things). But Inequality (24) can be applied just as well to operations on a wave function, such as multiplying it by x, or acting upon it with the momentum operator –(ih/2π) ∂x. With a little bit of calculus, we see that: [x, –(ih/2π) ∂x]ψ = = = –(ih/2π) x∂xψ + (ih/2π) ∂x(xψ) –(ih/2π) x∂xψ + (ih/2π) x∂xψ + (ih/2π) ψ (ih/2π) ψ ≥ = ≥ (1/2)|<ψ, (ih/2π) ψ>| (h/4π) |<ψ, ψ>| h/4π and so: ∆x ∆p ∆x ∆p (25) where we've used the fact that ψ is normalised, <ψ, ψ>=|ψ|2=1.

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