Superpositions

A generic (unknown) state |ψi, before measurement in a particular basis,

has the potential to be in either basis state. Suppose the probability am-

plitudes for measuring the state to be |↑i is a complex number α and that

for |↓i is β. We express this fact mathematically by writing |ψi as a linear

superposition

|ψi = α|↑i + β|↓i. (2.5)

Out of a beam of N electrons, a fraction |α|

2

would end up in the upper

beam and the fraction |β|

2

would be in the lower beam. Since all the electrons

emerge from the process, the total probability must be one:

|α|

2

+ |β|

2

= 1.

This unknown state |ψi, with the potential to be in either of the basis states,

is said to be in a superposition of the basis states.

The peculiarity of a superposition state is that until the system is mea-

sured, the state is not definite. It is difficult to visualize, with our classical

minds, an object that is in both basis states at once