*I think the answers given by Konrad Waldorf and Federico Poloni are fantastic. However, if you're just starting to learn quantum mechanics—especially from an older book, like the one you're using—you may want to think only about pure states (this can be done without loss of generality). In that case, this answer may be helpful for you.*

In my opinion, thinking about linear combinations of vectors as representing "superpositions" of the corresponding states is extremely misleading. It might suggest, for example, that if a photon in the state represented by $|H\rangle$ is guarnateed to pass through a horizontal polarizer, and a photon in the state represented by $|L\rangle$ is guaranteed to pass through a left circular polarizer, then a photon in the "superposition" state $|H\rangle + \sqrt{2}|L\rangle$ should have at least some chance of passing through a horizontal polarizer. However, by choosing the vectors $|H\rangle$ and $|L\rangle$ appropriately, you can set things up so that a photon in the state represented by $|H\rangle + \sqrt{2}|L\rangle$ will *never* pass through a horizontal polarizer.

Even when the concept of superposition isn't actively harmful, I find it totally unhelpful, and I would urge you to forget about it entirely. If you absolutely must have it, however, you may read on for a description of the only situation I know of in which "superposition" makes any kind of sense.

Suppose you have a quantum system with state space $\mathcal{H}$. You can think of each orthonormal basis for $\mathcal{H}$ as an abstract description of an experiment that could be done on the system; the basis vectors represent the possible outcomes of the experiment.

Say $|v_1\rangle, \ldots, |v_n\rangle$ is an orthonormal basis for $\mathcal{H}$, and
$$|\psi\rangle = c_1|v_1\rangle + \ldots + c_n|v_n\rangle$$
is a unit vector representing the state of the system. If you do the experiment described by the basis $|v_1\rangle, \ldots, |v_n\rangle$, you have probability $|c_k|^2$ of getting the outcome represented by $|v_k\rangle$.

Some people like to think of the state $|\psi\rangle$ as a "superposition" of the possible experimental outcomes represented by the basis vectors $|v_1\rangle, \ldots, |v_n\rangle$. Notice that if you change the coefficients $c_1, \ldots, c_n$, but keep their magnitudes the same, the state $|\psi\rangle$ will change, but the statistics of the experiment described by the basis $|v_1\rangle, \ldots, |v_n\rangle$ will stay the same. In other words, if
$$|\psi'\rangle = c'_1|v_1\rangle + \ldots + c'_n|v_n\rangle$$
is a superposition with $|c'_j| = |c_j|$ for all $j$, then **the states represented by $|\psi\rangle$ and $|\psi'\rangle$ cannot be distinguished using the experiment described by $|v_1\rangle, \ldots, |v_n\rangle$.**

However, consider another orthonormal basis $|w_1\rangle, \ldots, |w_n\rangle$, and write
$$|\psi\rangle = d_1|w_1\rangle + \ldots + d_n|w_n\rangle$$
$$|\psi'\rangle = d'_1|w_1\rangle + \ldots + d'_n|w_n\rangle.$$
In general, $|d_j|$ will not be equal to $|d'_j|$ for all $j$. In this case, **the states represented by $|\psi\rangle$ and $|\psi'\rangle$ can be distinguished using the experiment described by $|w_1\rangle, \ldots, |w_n\rangle$.**

So, if you have a "superposition" of orthonormal basis vectors, changing the coefficients without changing their magnitudes will not change the statistics of the experiment described by the basis you used, but it will generally change the statistics of the experiments described by most other bases. This is why changing the coefficients of a "superposition" generally gives you a representative of a different state.

I say "generally" because there is one exception: if you change your coefficients by multiplying them all by the same number, the statistics of all experiments will remain the same. This is why multiplying a vector by a number gives you a representative of the same state.

thesuperposition of them? $\endgroup$ – Michael Bächtold Dec 24 '12 at 15:204more comments