What does the notation $|x_1,x_2\rangle$ mean?

Question

I would like clarification on an equation in the paper "Free matter wave packet teleportation via cold-molecule dynamics", L. Fisch and G. Kurizki, Europhysics Letters 75 (2006), pp. 847-853, DOI: 10.1209/epl/i2006-10205-7.

The paper talks about entangling two particles translationally, meaning that two particles' position and momenta are correlated such that a precise measurement of particle 1 will cause particle 2's spread in momenta to be uncertain, vice versa.

So the equation is equation (2) in the paper,

$$\langle x_1, x_2 | \Psi \rangle= N e^{-\left({x_+}/{2\Delta x_+}\right)^2}N e^{-\left({x_-}/{2\Delta x_-}\right)^2}$$

where $x_+ = (x_1 + x_2)/2$, $x_- = x_1 - x_2$, and $N$ is a normalization constant.

I'm assuming that the $\Delta x_\pm$ are the standard deviations of $x_\pm$.

I've never seen bra-ket notation with "$\langle x_1,x_2|$" in it. This confuses me a lot! It doesn't make sense to have $x_1$ (comma) $x_2$. What the heck does this mean?

I am interpreting this as the expectation value of the positions of the two entangled particles where $|\Psi\rangle$ is the wave function of two translationally entangled particles. Can someone please help me?

Answer 1

For any quantum system with more than one coordinate - which may be a particle on more than one dimension, or several particles - the position kets need to be updated from the single-parameter $|x\rangle$ to accomodate the multiple coordinates $x_1,\ldots,x_n$, and are usually written $|x_1,\ldots,x_n\rangle$. The inner product you're asking about is simply the wavefunction corresponding to the quantum state $|\Psi\rangle$ in the (two-particle) position representation: $$\langle x_1,x_2|\Psi\rangle=\Psi(x_1,x_2).$$ Of course, it's a complex-valued function of its two arguments.

The pure kets are in fact tensor products: $|x_1,x_2\rangle=|x_1\rangle\otimes|x_2\rangle$. This is how multi-dimensional QM is usually built up: each degree of freedom has its Hilbert space $\mathcal{H}_j$, and the total Hilbert space is the tensor product $\mathcal{H}=\bigotimes_j \mathcal{H}_j$. This is perhaps best understood as the space of all (tensor) product wavefunctions, of the form $$\Psi(x_1,x_2)=\psi_1(x_1)\cdot\psi_2(x_2),$$ and all linear combinations of such, most of which cannot be written in that form, in which case the different degrees of freedom are called entangled. (Consider, for example, $\Psi(x_1,x_2)=N(x_1\cdot 1+1\cdot x_2)$.)