Hi all, I have a function for a particle, given by f(x) = e^ix and I wish to analyze it in a N-representation in a Hilbert space. What are the steps to a representation of f(x) in N-representation in Hilbert space?
Not quite sure how to interprate your question.
Maybe you mean something like you have exp(ikx), and you have a long box
with boundary conditions, so the values of k become discrete and then you get these N values for k?
You need k for dimensional reasons.
Sorry for any unclarity Juan, what I mean is how to represent a Hamiltonian and its solutions in a Hilbert space. The solution is i.e. e^ikx and the boundary conditions are 0 to 2pi.
Juan, I think this explains what I am wondering about:
https://physics.stackexchange.com/questions/246647/why-do-we-need-both-hamiltonian-and-hilbert-space-to-specify-a-quantum-system/246650
However, I am not sure which steps one should follow to prove the solutions to a Hamiltonian in a Hilbert space. So far, I have only used a Cartesian system with 2 dimensions, and got the eigenstate f(x) = e^ix and the interval of relevance is 0 to 2pi. Are there 3-4 steps one can stick to in order to interpret the Hamiltonian and its Cartesian solution in a Hilbert space for N particles?
Sergio,
unfortunately you express youself in a rather non-standard way. There is nothing like a 'solution to a Hamiltonian'. Also, it is not clear what a 'Cartesian system with 2 dimension' is to mean here. If you so far dealt with a single particle, there are in fact systematic ways to extend the treatment to N particles. Given my poor understanding of the system you have in mind, it makes not much sense for me to describe these ways at this stage. If, however, you would provide a reasonably detailed description of what you are doing, I would try to provide help with getting to a N-particle extension.
Sergio
In the quantum theory the wave function is an eigenstate of the Hamiltonian and can also be considered a vector in Hilbert space.
You can use k to label the different eigenstates for a problem that has a constant potential, either finite or infinite in size.
The representation of the Hamiltonian can then be something like
(k1 h k2) where h is the Hamiltonian.
This based on my current understanding of your question.
If you mean N particles, the wave function is
Psi(r1,r2,r3,....rN,t) a function of the N sets of coordinates.
The formalism of quantum mechanics is no more than linear algebra. As the Hamiltonien is an operator, it needs vectors to act on. The Hilbert space represents the system, and the Hamiltonian its dynamics.
For N particles, it depends on whether they are fermions of bosons. The wave function is a function of the configuration space, not of physical space. Thus there are N position x_i, and the wave function is antisymmetrised (fermions) or symmetrised (bosons) with respect to theses positions. For example for two fermions it is:
f(x_1,x_2)=f_1(x_1)f_2(x_2)-f_2(x_1)f_1(x_2)
For more fermions that is done using a determinant, the Slater determinant.
f(x_1,x_2,...) = det( (f_1(x_1),f_2(x_1), ...) ; (f_1(x_2),f_2(x_2),...) ; ... )
The usual way to analyze a function f(x) in a separable Hilbert space is using an appropriate countable set of orthonormal bra-ket base vectors |k>, f(k)
Thanks to all for this useful information. I have so far applied the Born interpretation, checked the overlap integral (which shows non-orthogonality) and developed a rationale towards distribution theory and Fourier integral for representing the wavefunction in k-space. The Hilbert space for N particles I have not used yet, as it would give the infinite Slater combination of N-particles with the given wavefunction. So, I am not sure how e^ikx can be represented using linear algebra in Hilbert space.
e^ikx is an eigenfunction of the operator i@/@x, that is, the linear momentum operator.
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