Welcome to Gaia! ::

Most questions central in mechanics (classical or quantum mechanical) boil down to trying to measure either position or momentum. We make these entities (pseudo-)vectors and content ourselves in measuring the components of these vectors in some prescribed coordinate system. Now we know if something starts off at one position with one momentum and let it evolve in time, external forces can alter both of these quantities.

How we handle this evolution differs depending on whether we're using quantum mechanical treatments or classical formalisms. Classically, we can always just write down the position and momentum and let their values evolve in a determined way. Quantum mechanically, we give the system states in Hilbert space that correspond to having different values of energy and momentum.

Recall how I mentioned we choose a coordinate system and measure the components of the vectors? Well, we do the same in quantum mechanics. Pick a basis for Hilbert space, and measure the components of the system's state in that basis. Consider the set of state vectors that correspond to each possible position something can have. This turns out to form a basis for Hilbert. Moreover, since position is part of a continuous set, the set of position states are also continuous. Thus, instead of a discrete number of components, there's an infinite number, labeled continuously by the position variable.

Consider an electron. The function that tells us the component of the electron's state along a particular position basis state is called the the wave function. It's complex. So if you find it's magnitude and square it, it turns out to be a probability density.

What's this mean? Let P(x) be the square of the magnitue I mentioned. If we center a small box of volume dV at the position x then the probability of finding the electron in that box is P(x)*dV.

I'll post more later, but for now: dinner! YUM!

Hope it helps.

Any time! ^_^

I started getting a little rushed there since I'm home for the summer and my rents and I were heading out to dinner. So I'll try to elaborate a lil' bit.

The distinction between quantum mechanical treatments and classical treatments probably didn't seem all that striking the way I explained it. I said we deal directly with the vectors themselves in classical treatments but just represent these vectors using states in a Hilbert space. This sounds no different than essentially dealing with an object versus dealing with a picture of it.

The distinction is more striking than that, really. The states don't necessarily correspond to a particle having an exact position or momentum. The set of states that correspond to exact momenta or positions form a basis for Hilbert space and the square of the magnitude of the components of the particle's actual state along each of these basis states tells us the probability (density) that we'll find the particle in that state when we measure the position or momentum. So the position and momentum doesn't evolve deterministically, but this Hilbert state does and the representation of this state in the position basis is the famous wave function. Since its magnitude squared is a probability density you here about probability waves.

Hopefully that might make things a little more clear.

They can't....at least up to spin they can't. Label the two electrons as A and B and consider a 2 state configuration where an electron can either be in state 1 or state 2. Then the state corresponding to electron A being in state one can be labeled as |A,1>. The composite state for the two electrons is then
|Y>=(1/sqrt(2))*(|A,1>|B,2>-|A,2>|B,1>). As you can see, if |A,1>=|A,2>, by symmetry |Y>=0, which means the system can't exist. Therefore, no two entangled electrons can be in the same state. This is an application of Pauli's exclusion principle.

In general, when dealing with entangled particles, you can't really discuss a single particle state. Let's say we have one particle in a state |A,1> and another particle in a state |B,2>. Then the composite state is |Y>=|A,1>|B,2>. This is separable in the sense that the composite state is the tensor product of two single particle states. We can therefore talk about individual states and manipulate them independently.

However, let's say the particles are entangled. Then the composite state would look more along the lines of
|Y>=(1/sqrt(2))*(|A,1>|B,2>+/-|A,2>|B,1>). This can't be written as the tensor product of two individual single particle states, so we can't manipulate each particle's state individually and are forced to deal with the composite state. So for bosons (the + case of the above expression) the states can be the same, but don't have to be and we can't talk about independent individual particle states (hence why such states are called 'entangled').

Entanglement is ordinarily addressed in a 2nd semester course on quantum mechanics intended for 3rd year B.S. majors at my university. It was more thoroughly examined in the corresponding 2nd semester graduate version of the course.