Editing Quantum superposition (section)

==Theory==

=== General formalism ===
Any quantum state can be expanded as a sum or superposition of the eigenstates of an Hermitian operator, like the Hamiltonian, because the eigenstates form a complete basis:
:<math>
 |\alpha\rangle = \sum_n c_n |n\rangle,
</math>

where <math>|n\rangle</math> are the energy eigenstates of the Hamiltonian. For continuous variables like position eigenstates, <math>|x\rangle</math>:

:<math>
 |\alpha \rangle = \int dx' |x'\rangle \langle x'|\alpha \rangle,
</math>

where <math>\phi_\alpha(x) = \langle x| \alpha \rangle</math> is the projection of the state into the <math>|x\rangle</math> basis and is called the wave function of the particle. In both instances we notice that <math>|\alpha\rangle</math> can be expanded as a superposition of an infinite number of basis states.

===Example===
Given the Schrödinger equation

: <math>\hat H |n\rangle = E_n |n\rangle, </math>

where <math>|n\rangle</math> indexes the set of eigenstates of the Hamiltonian with energy eigenvalues <math>E_n,</math> we see immediately that

: <math>\hat H\big(|n\rangle + |n'\rangle\big) = E_n |n\rangle + E_{n'} |n'\rangle,</math>

where

: <math>|\Psi\rangle = |n\rangle + |n'\rangle</math>

is a solution of the Schrödinger equation but is not generally an eigenstate because <math>E_n</math> and <math>E_{n'}</math> are not generally equal. We say that <math>|\Psi\rangle</math> is made up of a superposition of energy eigenstates. Now consider the more concrete case of an [[electron]] that has either [[Spin (physics)|spin]] up or down. We now index the eigenstates with the [[spinor]]s in the <math>\hat z</math> basis:

: <math>|\Psi\rangle = c_1 |{\uparrow}\rangle + c_2 |{\downarrow}\rangle,</math>

where <math>|{\uparrow}\rangle</math> and <math>|{\downarrow}\rangle</math> denote spin-up and spin-down states respectively. As previously discussed, the magnitudes of the complex coefficients give the probability of finding the electron in either definite spin state:

: <math> P\big(|{\uparrow}\rangle\big) = |c_1|^2,</math>
: <math> P\big(|{\downarrow}\rangle\big) = |c_2|^2,</math>
: <math> P_\text{total} = P\big(|{\uparrow}\rangle\big) + P\big(|{\downarrow}\rangle\big) = |c_1|^2 + |c_2|^2 = 1,</math>

where the probability of finding the particle with either spin up or down is normalized to 1. Notice that <math>c_1</math> and <math>c_2</math> are complex numbers, so that

: <math>|\Psi\rangle = \frac{3}{5} i |{\uparrow}\rangle + \frac{4}{5} |{\downarrow}\rangle.</math>

is an example of an allowed state. We now get

: <math>P\big(|{\uparrow}\rangle\big) = \left|\frac{3i}{5}\right|^2 = \frac{9}{25},</math>
: <math>P\big(|{\downarrow}\rangle\big) = \left|\frac{4}{5}\right|^2 = \frac{16}{25},</math>
: <math>P_\text{total} = P\big(|{\uparrow}\rangle\big) + P\big(|{\downarrow}\rangle\big) = \frac{9}{25} + \frac{16}{25} = 1.</math>

If we consider a qubit with both position and spin, the state is a superposition of all possibilities for both:

:<math>
 \Psi = \psi_+(x) \otimes |{\uparrow}\rangle + \psi_-(x) \otimes |{\downarrow}\rangle,
</math>

where we have a general state <math>\Psi</math> is the sum of the [[tensor products]] of the position space wave functions and spinors.