Linked Questions

2
votes
1answer
66 views

What is the scope of the term 'normalisation'?

When we 'normalise' the wavefunction we put in an appropriate coefficient so that the wavefunction can act as a probability distribution. However, when I considered the eignefunctions of the momentum ...
0
votes
1answer
144 views

Eigenfunctions of observables

Are eigenfunctions of observables solutions to the time-dependent Schrödinger equation? Or is this not necessarily the case? From what I had been reading they are not necessarily solutions to ...
0
votes
1answer
215 views

Probability of finding an energy state of a non-normalisable wave-function

Suppose, say, I have the following wave function It represents the wave function of a free particle. I would want to calculate the probability of finding the particle with energy ħk and energy 2ħk. ...
2
votes
2answers
481 views

Quantum field theory for the gifted amateur: problem 2.4

I am trying to do problem 2.4 in the book "Quantum field theory for the gifted amateur". I have a math background but little training in physics. I am asked to use the identity $$\langle x \mid p \...
2
votes
1answer
224 views

Bras and kets of continuous spectrum

Does anyone know why in quantum mechanics the second statement is always true? "When the spectrum of an operator $A$ has a continuous part, we associate a bra $\langle a|$ and a ket $|a \rangle$ ...
2
votes
1answer
328 views

Protocol for solving time independent Schrodinger equation

Just a short question about the protocol for solving the time-independent Schrodinger equation for different potentials and the reasons for accepting and rejecting solutions. Take for example the ...
0
votes
1answer
218 views

Is it possible to decompose into eigenstates of Dirac Hamiltonian?

If we have the Hilbert space $\mathcal H = L^2(\mathbb R^3, \mathbb C^4)$ and a Hamiltonian: $$H=\gamma^i p_i + m \gamma^0$$ where $\gamma^i$ are matrices and $\{\gamma^i,\gamma^j\}=\delta^{ij}$. A ...
12
votes
2answers
440 views

Can a normalizable function *always* be decompose into the discrete Hydrogen spectrum?

This question has been bothering me for a while now: can one reconstruct an arbitrary (normalizable) function $\phi(\mathbf r)$ in $\mathbb R^3$, with only the (discrete) set of Hydrogen ...
13
votes
2answers
928 views

Must bounded operators have normalisable eigenfunctions and discrete eigenvalues?

When we have bound states, to my knowledge, we have states that are normalisable and a discrete energy spectrum. However, in the case of scattering states that have a continuous energy spectrum, the ...
12
votes
3answers
3k views

What really is a Dirac delta function?

Yesterday a friend asked me what a Dirac delta function really is. I tried to explain it but eventually confused myself. It seems that a Dirac delta is defined as a function that satisfies these ...
2
votes
1answer
844 views

Continuous spectrum of hydrogen atom

I wonder if there is a nice treatment of the continuous spectrum of hydrogen atom in the physics literature--showing how the spectrum decomposition looks and how to derive it.
2
votes
2answers
1k views

Does a free electron, one that's not either in an atom or a wire, have an associated wave-function?

Would a free electron, one that's not either in an atom or moving through a wire, but moving through empty space on its own, have an associated wave-function? Or, is an electron described as a wave-...
8
votes
2answers
1k views

How to guarantee square integrable solutions to time-independent Schrödinger's equation?

Given the time-independent Schrödinger’s equation in one dimension $$H\psi = E\psi$$ what restrictions can we place on V(x) (inside the hamiltonian) and E to guarantee that the solutions won't have ...
1
vote
2answers
384 views

About the orthogonality of the Hamiltonian eigenstates for the the continuous energy spectrum

I would like first to describe a strange case that I encountered. $ \ \ - $ I solved the Schrodinger equation with a potential barrier (a potential well limited by a finite height wall which decrease ...
9
votes
2answers
1k views

Infinite dimensional vector spaces vs. the dual space

I just happened across this over on Math Overflow. It references the following theorem from linear algebra: A vector space has the same dimension as its dual if and only if it is finite dimensional....
3
votes
2answers
696 views

Is a wave packet physically realizable as a Fourier series?

In QM a wave packet is modeled as an infinite, or almost infinite, Fourier series, and the Fourier transform provides a transformation between momentum space and position space. To what extent is ...
10
votes
5answers
7k views

How does the research in theoretical physics differ from mathematics [closed]

I would like to know what is the difference between research in theoretical physics and pure mathematics. In particular, what does a theoretical physicist actually do all day long for his research? In ...
1
vote
1answer
594 views

In quantum mechanics, why position and momentum are related by Fourier Transformation (only)? [duplicate]

We know that if we take Fourier transform of momentum we go to position space. But why Fourier transform only.
9
votes
2answers
491 views

Does the general topology of Minkowski space-time change under a Lorentz transformation?

Does the general topology of Minkowski space-time change under a Lorentz transformation? Open balls in $\mathbb{R}^{4}$ (with the standard topology) are not invariant under Lorentz transformations. ...
3
votes
3answers
897 views

How to understand holography and hologram

I've spent some time reading wiki etc. What I get now is that apart from the normal light amplitude information, holograms also record the phase information of light. But this is so difficult for me ...
9
votes
2answers
2k views

Does the wave function always asymptotically approach zero?

I'm new to quantum physics (and to this site), so please bear with me. I know that quantum mechanics allows particles to appear in regions that are classically forbidden; for example, an electron ...
2
votes
1answer
647 views

Heisenberg relation

Given that $A(k)=\frac{N}{k^2+\alpha^2}$, show that $\Delta k \Delta x >1$. Considering the above example, according to my textbook, it is written that I must square the above function and ...
4
votes
1answer
468 views

Distributions (e.g., Dirac Delta): confused and unhappy [closed]

I am sorry that the following set of questions is very fuzzy and ill informed. I am a trained mathematician and now studying an undergraduate theoretical physics course. We use distributions. I have ...
1
vote
4answers
1k views

Why complex functions for explaining wave particle duality?

I have this very bad habit of going to the scratch, discarding all the developments of a theory and worldly knowledge, and ask some fundamental (mostly stupid and naive, as some may say) questions as ...
7
votes
2answers
2k views

Proving that $i\hbar\frac{\partial}{\partial \mathbf{p}}$ is the operator of $\mathbf{x}$ in momentum space

How can I prove that $i\hbar\frac{\partial}{\partial \mathbf{p}}$ is the operator of $\mathbf{x}$ in momentum space?
4
votes
2answers
720 views

What restrictions on time boundary conditions does it have to use Fourier transform to solve wave equation?

The wave equation can be solved using Fourier transform, by assuming a solution of the form of $$\mathbf{E}(x,y,z,t)~=~\mathbf{E}(x,y,z)e^{j\omega t}$$ and then reducing the equation to the Helmholtz ...
6
votes
1answer
2k views

What physical significance has the Heisenberg Group?

I read that the canonical commutation relation between momentum and position can be seen as the Lie Algebra of the Heisenberg group. While I get why the commutation relations of momentum and momentum, ...
11
votes
2answers
4k views

Why are eigenfunctions which correspond to discrete/continuous eigenvalue spectra guaranteed to be normalizable/non-normalizable?

These facts are taken for granted in a QM text I read. The purportedly guaranteed non-normalizability of eigenfunctions which correspond to a continuous eigenvalue spectrum is only partly justified by ...
36
votes
2answers
5k views

Rigged Hilbert space and QM

Are there any comprehensive texts that discuss QM using the notion of rigged Hilbert spaces? It would be nice if there were a text that went through the standard QM examples using this structure.