Applies also to pre-Hilbert spaces, rigged Hilbert spaces, and spaces with negative norm or zero-norm states.

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8
votes
4answers
297 views

Is it possible to reconstruct the Hamiltonian from knowledge of its ground state wave function?

Is it possible to "construct" the Hamiltonian of a system if its ground state wave function (or functional) is known? I understand one should not expect this to be generically true since the ...
0
votes
2answers
59 views

Completeness relation for coherent states of the quantum harmonic oscillator

For the Quantum harmonic oscillator with energy eigenstates $|n\rangle$ one defines a coherent state for every complex number $z$ by setting (note that the normalization varies across the literature) $...
1
vote
0answers
40 views

Trouble understanding Nielsen & Chuang exercise

I am probably just stuck on something very simple, but I'm having trouble understanding a premise of Exercise 10.40 in Nielsen & Chuang. The full details of the exercise are not important for my ...
0
votes
1answer
33 views

Representation of spin operators in a two electrons system

I've studied that the spin space of an electron is a two-dimensions Hilbert space. A possible representation of this space can be constructed defining: $$\chi_+ = \begin{pmatrix}1 \\ 0 \end{pmatrix} \...
1
vote
1answer
56 views

$\frac{1}{\sqrt{2}}(1 + i)|0\rangle$ on the Bloch sphere

By definition, a quantum state can be expressed as $$|\psi\rangle = a |0\rangle+b |1\rangle.$$ Here, $a, b\in\mathbb{C}$ and $|a|^2 + |b|^2 = 1$. Now, I would like to take $a = \frac{1}{\sqrt{2}}(1 +...
2
votes
0answers
106 views

Linear combination of eigenstates problem [closed]

Let's say that we have a system such that $$\Psi(x,0)=\frac{\sqrt3}{2}\phi_1(x)+\frac12\phi_2(x)$$ where both $\phi(x)$ are eigenfunctions of the Hamiltonian operator. I want to find $\Psi (x,t)$ ...
2
votes
2answers
147 views

Eigenvalues of Hermitian operators are real and the dependence/independence of boundary conditions

Without reproducing proofs: Eigenvalues of a Hermitian operator are real (proof does not rely on the boundary conditions). The momentum operator is Hermitian (proof does not rely on the boundary ...
0
votes
1answer
41 views

How to understand the electric-field operator in quantum optics?

I know the positive field operator $\mathbf{E}^{+}$ is actually an annihilation operator $a$ while the negative field $\mathbf{E}^{-}$ is a creation operator $a^{+}$. I also learned that the ...
29
votes
5answers
3k views

Why do we need infinite-dimensional Hilbert spaces in physics?

I am looking for a simple way to understand why do we need infinite-dimensional Hilbert spaces in physics, and when exactly do they become neccessary: in classical, quantum, or relativistic quantum ...
-3
votes
2answers
68 views

In handling two 1/2-spin particles, why is there only one singlet state? [closed]

Why is $\left|\uparrow\uparrow\right\rangle +\left|\downarrow\downarrow\right\rangle$ not discussed, despite having a total spin s = 0?
1
vote
1answer
38 views

π , σ - atomic transitions with respect to a quantization axis

In the absence of a magnetic field, how does one physically (i.e perhaps in a thought expt) access delta_m = 0 or +-1 transitions since (as I understand it) the choice of quantization axis is ...
-1
votes
0answers
18 views

Normalization operator

Is it possible to define an $\widehat N:\mathscr{H\to H}$ such that $\forall\left|\psi\right>\in\mathscr{H}: \widehat N\left|\psi\right>=\frac1{\sqrt{\left<\psi\middle|\psi\right>}}\left|\...
2
votes
1answer
113 views

Why are general wave functions expressed in terms of energy eigenfunctions?

I have read that the eigenfunctions of any hermitian operator can be used as a basis to express any function, but I have only ever really seen the eigenfunctions of the Hamiltonian used. Why is this? ...
2
votes
1answer
57 views

Different postulates and statistical interpreations of quantum mechanics

Hi I have a query about the difference of two aspects of the statistical interpretation of quantum mechanics given in the popular introductory quantum mechanics books "Introduction to Quantum ...
4
votes
1answer
89 views

Understanding the relationship between Phase Space Distributions (Wigner vs Glauber-Sudarshan P vs Husimi Q)

I am moving into a new field and after thorough literature research need help appreciating what is out there. In the continuos variable formulation of optical state space. (Quantum mechanical/Optical)...
5
votes
1answer
142 views

What is the difference between active and passive transformations in Quantum Mechanics?

I am trying to understand what each transformation means and what their differences are but many books that don't state which transformation they are referring to make it a bit confusing to understand ...
12
votes
2answers
985 views

How should Dirac notation be understood?

If vectors $|\vec{r}⟩$ and $|\vec{p}⟩$ are defined as $$ \hat{\vec{r}} |\vec{r}⟩ = \vec{r} |\vec{r}⟩ \\ \hat{\vec{p}} |\vec{p}⟩ = \vec{p} |\vec{p}⟩ $$ then one can see that products like $$ ⟨\vec{...
5
votes
1answer
183 views

BRST quantization and norm

States with definite ghost number have zero norm (since ghost number is anti-hermitian and has real eigenvalues). E.G. when quantizing relativistic point particle, physical spectrum turns out to ...
12
votes
1answer
330 views

Shape of the state space under different tensor products

I am currently studying generalized probabilistic theories. Let me roughly recall how such a theory looks like (you can skip this and go to "My question" if you are familiar with this). Recall: In a ...
15
votes
4answers
718 views

Curvature of Hilbert space

That may appear as a dumb question, but: Does Hilbert space have curvature, or is it a flat space? How and why?
-2
votes
0answers
54 views

Expectation value of an Observable and Eigenstates

I am learning about Quantum Mechanics at the moment and I was wondering about Eigenfunctions and Observables. The question I would like to ask is, If a wavefunction is not an eigenstate of an ...
2
votes
3answers
61 views

Two qubits system in polar co-ordinates

I know that I can write a single qubit state in terms of polar co-ordinates $(r,\theta,\phi)$ on a Bloch sphere. \begin{equation} \rho = \begin{pmatrix} \frac{1+r \cos\theta}{2} &\frac{r \exp(-i\...
12
votes
1answer
249 views

Significance of the exception to Gleason's Theorem when n = 2

Gleason's Theorem famously asserts that (appropriately defined) measures on the lattice of a complex Hilbert space can be implemented by density operators via the trace operation, except in the case ...
1
vote
0answers
41 views

Are Fock spaces just a special type of tensor algebra?

Are Fock spaces just a special type of tensor algebra? The definitions I am using: http://en.wikipedia.org/wiki/Fock_space http://en.wikipedia.org/wiki/Tensor_algebra From what I can tell, the ...
4
votes
1answer
152 views

Coherent states and completeness

Consider one possible definition of a Gaussian (coherent) state in the position representation $$ \langle r | \psi(r_i,p_i) \rangle = \left( \frac{ 2 \gamma}{\pi} \right)^{\frac{1}{4}} \exp \left[ -\...
0
votes
2answers
154 views

General formula for expanding wave function in terms of orthogonal states?

Given a wave function $\psi(x) = \langle \psi | x \rangle$. It can be expanded in terms of orthogonal states: $$ \langle \psi | x \rangle = \sum_n \langle \psi | n \rangle \langle n |x \rangle $$ ...
1
vote
1answer
83 views

What does coherent superposition mean?

There is only one coherent state: $$|\alpha\rangle=e^{-\frac{|\alpha|^2}{2}}\sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}}|n\rangle $$ Also, a pure state does not mean a coherent state. But when does ...
0
votes
2answers
48 views

In quantum double slit experiment what is the state vector or density matrix of the electron after the electron passes through the two slits?

Also I would like to confirm my thinking on quantum double slit experiment. Before it passes through the two slits (slit 1 and slit 2), is the electron state vector $\frac{1}{\sqrt{2}}\left(\left|...
0
votes
0answers
35 views

Why Does the Dirac delta Function Fix the Normalization of the Basis Vectors in Infinite Dimensions? [duplicate]

On page 60 of Shankar's intro to QM at the very bottom he says that the Dirac delta function fixes the normalization of the basis vectors with an infinite amount of dimensions. I don't understand why ...
4
votes
1answer
86 views

Is the usage of the Fock space a postulate in QFT?

In this question, when I write Fock space, I mean "the direct sum of the symmetric or antisymmetric tensors in the tensor powers of a single-particle Hilbert space H", as it is described by Wikipedia. ...
2
votes
2answers
356 views

Commuting operators and Direct product spaces

Under what conditions is the common eigenspace of two commuting hermitian operators isomorphic to the direct product of their individual eigenspaces? When can an eigenket $|\lambda$1$\lambda$2$\rangle$...
2
votes
1answer
75 views

How to understand permutations of particles in Quantum Mechanics?

I'm studying identical particles in Quantum Mechanics and I'm having a hard time to understand the idea of permutations of particles from a mathematical standpoint. From one intuitive point of view ...
0
votes
1answer
32 views

Normalisation of angular wave function: particle in a circular box

For a particle in a circular box (with radius $R$) with zero potential inside the circle and infinitely high potential outside of the circle, the Schrödinger equation in polar coordinates is: $$-\...
0
votes
1answer
171 views

Expectation value of total angular momentum $\langle J \rangle$

[I am working with Griffiths Introduction to Quantum Mechanics, 3rd Edition. My problem is general but if you want to look I am reading from ch 4.1 in which the weak-field Zeeman Effect is being ...
0
votes
1answer
46 views

Want to measure entanglement of the state [closed]

Good day, I want to measure the state with concurrence and negativity. I do local unitary transformation with represented by $U\in SU(4)$ (Lie group). After the transformation (rotation of angle) ...
1
vote
1answer
92 views

Probability in QM: derivation or interpretation? [duplicate]

It is known that coordinates $C_k\in\mathbb{C}$ of the QM-state vectors $|\psi\rangle$ has an interpretation as probability weights $p_k$ in the whole state through the formula like $|C_k|^2=p_k$. We ...
7
votes
2answers
113 views

Properties of spectrum of a self-adjoint operator on a separable Hilbert space

So, if I understand it correctly, the spectrum of a self-adjoint operator on a Hilbert space $H$ consists of two parts: $ \newcommand{\ket}[1]{\,\lvert{#1}\rangle} \newcommand{\op}[1]{\hat{#1}} $ ...
6
votes
3answers
515 views

Hilbert space of a quantum system

The first postulate of Quantum Mechanics as I've learned can be stated as: The states of a quantum system can be described by vectors in a Hilbert space. I've seem also some people also ...
0
votes
1answer
48 views

Why Does there Have to be Linearity in Ket and Skew Symmetry?

I'm reading Shankar's "Principles of Quantum Mechanics," and on page 8 he states that one axiom in Dirac notation is linearity in ket, and because they are also skew symmetric there is anti-linearity ...
2
votes
1answer
235 views

What is many-body bound state?

Bound state by definition is a state when particles are bounded together, so then "many-body bound state" would be bound state for a system of many bodies. Then I have several puzzles: is the state ...
2
votes
0answers
58 views

Rigorous way of box normalisation

This is follow up from an answer to my previous question about unitarity in rigged Hilbert space. As it turns out, that there is no idea of unitarity in rigged Hilbert space (hence no meaningful QM ...
2
votes
1answer
60 views

Norm preserving Unitary operators in Rigged Hilbert space

If we take the free particle Hamiltonian, the eigenvectors (or eigenfunctions), say in position representation, are like $e^{ikx}$. Now these eigenfunctions are non-normalisable,so they don't belong ...
13
votes
2answers
547 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 ...
2
votes
2answers
111 views

Does Unitary operator take a pure state to a pure state or can it take a pure state to a mixed state? [closed]

Does Unitary operator take a pure state to a pure state or can it take a pure state to a mixed state? I think so but why? I assume the Unitary operator acts on a pure state only.
3
votes
2answers
93 views

What kets represent on QFT?

In Quantum Mechanics kets are used to represent states of a system. This is indeed well written in the first postulate of Quantum Mechanics which states that to describe a quantum system we use a ...
1
vote
0answers
52 views

How can we justify identifying the Dirac delta function with the eigenfunction of position? [duplicate]

I can think of at least two different ways to understand eigenfunctions of operators in quantum mechanics. But neither one seems to provide a good explanation for why we take the position-basis ...
3
votes
3answers
146 views

Hilbert Space axiom in QM

My question is about the standard axiom on Hilbert's space in orthodoxal QM. It seems that this axiom appeares actually as an external pure mathematical axiom in all textbooks. Say, Mackey introduces ...
0
votes
2answers
81 views

How can I show that $\mathrm{Tr}\left( f(G^\dagger G)\right)=\mathrm{Tr}\left( f(G G^\dagger)\right)$?

I'm slightly stuck on the following question: Prove that: $\mathrm{Tr}\left( f(G^\dagger G)\right)=\mathrm{Tr}\left( f(G G^\dagger)\right)$ where $G$ is any operator. Using the definition of the ...
11
votes
3answers
427 views

Can superpositions of baryons with different electric charge and strangeness exist?

I am trying to find out whether the following baryons can exist: $$ |X\rangle = \frac{|u u u\rangle + |d d d\rangle + |s s s\rangle}{\sqrt{3}} $$ $$ |Y\rangle = \frac{|u u u\rangle + |d d d\rangle - ...
1
vote
1answer
36 views

Dirac notation - trace of product of (bipartite) density matrices

I'm getting confused by the Dirac notation. Suppose I have the following two objects. $$\rho = \sum_k p_k (\rho_A \otimes \rho_B) = \sum_k p_k |k \rangle \langle k | \otimes |k\rangle \langle k | ,$$...