A lot of confusion in Quantum Mechanics is the result from not being able to differentiate between the real world and the Hilbert Space. Vectors in real space – like velocities, accelerations, forces, etc. – are objects one can actually measure in the real world. On the other hand, quantum states are represented by vectors (more precisely by rays) in a Hilbert space, but these are NOT subjects of measurement. What we measure for a quantum system are probabilities, and those vectors in that Hilbert space are useful mathematical tools to calculate those probabilities.
Illustration
Suppose we have a beam of electrons flowing from right to left:
Notice this is a thought experiment as we really don’t know in what direction the spin of each individual electron points. We can safely say that these directions are at random. Now physicists are interested in measuring these spins. So I need some kind of apparatus, and the good news is that there exists one – a magnetic field. Trouble is that these electrons, with their spin, are tiny magnets, and we know that magnets placed in a magnetic field will align (or anti-align) with the magnetic field. Suppose I place the magnetic field along a certain direction, say the z-axis. Now let’s look at one electron as it approaches the magnetic field.
When that electron penetrates the magnetic field, it will align its spin such that its z-component will yield the value of +ℏ/2 along the z-axis, a spin up, which can be represented as:
Here’s another electron about to penetrate the magnetic field:
This time it will anti-align with the magnetic field, with a spin value of -ℏ/2, a spin down.
On the whole, 50% of the electrons will align with the magnetic field (spin =+ℏ/2, or up), and 50% will anti-align (spin = -ℏ/2, or down).
Comments
(1) Note that before the measurement, the spin of an electron can be in any direction. Passing the electron through the magnetic field forces the electron to change its spin orientation such that it either aligns or anti-aligns with its z-component to be ± ℏ/2. This is what distinguishes quantum physics from classical physics: the act of measuring a quantity will disturb the system.
(2) The other components of the spin are indeterminate: if I were to pass these electrons into another magnetic field, say aligned with the x-axis, again I will find that 50% of the electrons will align with the magnetic field (spin = +ℏ/2), and 50% will anti-align (spin = -ℏ/2), this time along the x-axis. I will no longer know what the spin along the z-axis is.
(3) One way to mathematically represent this quantum system (read, the wave function) is this:
| ψ> = 1/(2^{½}) (| ↑ > + | ↓ >).
Now this is called a superposition of two quantum states, the up and down states. Note that if I want to calculate the probability that the electron has a spin up, I take the product of the vector | ↑ > with the wave function | ψ>, and square that.
P = |< ↑| ψ >|^{2}
= 1/2 [< ↑ | { | ↑ > + | ↓ > }]^{2}
=1/2 [{ < ↑ | ↑ > + < ↑ | ↓ > }]^{2}
Using the orthogonality condition, < ↑ | ↑ > = 1 and < ↑ | ↓ > =0, we get,
P=1/2, or 50%, which is what is observed in the lab.
(4) Now here comes the real crunch. Writing | ψ> = 1/(2^{½}) (| ↑ > + | ↓ >) is called a superposition but it’s not meant to mean that the electron “lives” simultaneously in two states and can’t make up its “mind” in which one it wants to live. Those states do not represent ordinary vectors of real objects - like velocities, acceleration, forces, which was a crucial point that was mentioned above. If it were the case, then since these two vectors are equal in magnitude and opposite in direction I would be able to claim,
| ↑ > = (-1) | ↓ >,
and the orthogonality condition would no longer hold, and P would not equal to 50% - actually it would turn out to be zero!!! What needs to be reminded is that the two vectors, | ↑ > and | ↓ > represent possible states, and the beauty of it all is that they form a complete set of orthogonal unit vectors, in an abstract space called the Hilbert space, which provides a powerful method of calculating probabilities.
Appendix
A word on semantics: note that I used the word "apparatus" when that word description is NOT needed. For instance at the LHC, one thinks of two beam interacting (colliding), and not as one beam interacting with an apparatus - the second beam. Similarly, the beam of electrons described above are interacting with a magnetic field (the "apparatus"). Hence, the whole concept of "wave function collapse" is totally unnecessary. The so-called measurement between a microscopic system and a macroscopic system is illusive as it never happens, it is always a microscopic system interacting with another microscopic system. And the wave function cannot collapse as it is not a function of a real wave. Also, there is no need of hidden variables or "beables". QM can do very well without this extra baggage.