Questions about the formalism of Quantum Mechanics I have to do a presentation on this. I'm not expected to do something really detailed, but I'm not understanding the mathematical formalism. I would like to receive general answers to these questions:


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*What is the configuration space $R^{3N}$? What does this mean? I've seen definitions on Wikipedia and they are way, way beyond me.

*The wave function is evaluated with certain values for $q$ and $q$ is $(\mathbf{q_1}...,\mathbf{q_N})$ what is this quantity? Is it like a vector of vectors?

*The hamiltonian here is $-\sum_{k=1}^{N}\frac{\hbar}{2m_k}\nabla_k^2+V$ How do you evaluate $\nabla_k$ of a wavefunction with the strange argument defined above? Namely $\Psi=\Psi(q,t)$

*How is the potential $V$ evaluated? What are its arguments?

*The guiding equation is $\frac{d\mathbf{Q_k}}{dt}=\frac{\hbar}{m}{Im}\frac{\nabla_k \Psi}{\Psi}(Q(t))$. What is the imaginary part of such a term which seems like a vector to me? I only know taking imaginary parts of complex numbers, not vector quantities.

*What is the difference between $q(t)$ and $Q(t)$ ?
Note that I've used the notation and symbols presented in many different sources. I assume you'll know what I mean with them.
 A: First I want to point out that most of these questions do not bring up issues specific to Bohmian mechanics. That's not a criticism, I'm just pointing out that these notations and concepts are already employed in standard quantum mechanics, or even in classical mechanics. 
I am going to answer this a little casually, and then make my answer "community wiki", in the hope that someone with more time will perhaps improve the exposition. 
The position in space of a single particle depends on three coordinates, e.g. (x,y,z). Mathematically, the position is an element of R^3, the real numbers "cubed". The coordinates for N particles consist of N sets of 3 real numbers, thus form an element of R^3N, 3N copies of the real numbers. 
One of the little q-vectors - let's say q_k for some number k - stands for a point in the kth copy of R^3. It is a possible position of particle k. We could write it out as (x_k,y_k,z_k). So the "vector of vectors" is really just (x1,y1,z1,x2,y2,z2,x3,y,3,z3,...,xN,yN,zN). It's a set of 3N variables which together range across all the possible positions of N particles. 
Ψ is a function of these 3N spatial coordinates; it could also be written as Ψ(x1,y1,z1,x2,y2,z2,x3,y,3,z3,...,xN,yN,zN). 
∇ in 3-space (R^3) is a 3-vector of directional derivatives. ∇ in R^3N is, to use your phrase again, a "vector of vectors" - a set of N such 3-vectors - a vector in 3N-space formed of partial derivatives with respect to the 3N variables. You are making a vector out of the derivatives of Ψ with respect to x1,y1,z1,x2,y2,z2,... It's exactly analogous to the usual three-dimensional ∇. 
∇_k is the directional derivative of Ψ just with respect to x_k, y_k, z_k. (∇_k)^2, when evaluated, would be a scalar resulting from taking the dot product (∇_k)Ψ . (∇_k)Ψ
V is some function of the q's, the N position vectors. To evaluate, you multiply it by Ψ. 
The difference between q and Q is like the difference between the x axis, and a particular value of X. If we forget for a moment about multiple particles, and even about three dimensions of space, and just consider the Bohmian mechanics of a single particle moving back and forth along a line, which we will label "x"... There are two entities in the theory: a wavefunction which stretches all the way along the line, and a particle which is always located a specific point on the line, and which is moved around by the wavefunction. Suppose we call this wavefunction "Ψ1", just to emphasize its one-dimensionality. As a function, Ψ1 depends just on the one variable "x", we would write it as Ψ1(x); but Ψ1 always has a value at every possible x, from -infinity to +infinity. 
The particle's position we are designating by X. It's always some particular number in that range. So the difference between x and X, is that x is an index running along the full length of the wavefunction, it labels the points on the wavefunction so you can talk about them and say things like Ψ1(-2) = 0.3. Whereas X is a single number, the current position of the particle. 
Returning to the N-particle case, at any moment the wavefunction is a 3N-dimensional waveform, stretching throughout the 3N-dimensional "space"; q is the set of coordinates for that space, Q is a particular location in that space. 
For "Im", you're just taking the imaginary part of each component separately. 
