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

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2
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2answers
90 views

Separability of a Hilbert space and its implications for the formalism of QM

In the text I'm using for QM, one of the properties listed for Hilbert space that is a mystery to me is the property that it is separable. Quoted from text (N. Zettili: Quantum Mechanics: Concepts and ...
1
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3answers
76 views

State vector vs density operator

We formulate quantum mechanics using language of state vectors. One alternative formulation is possible using density operator or density matrix. Why we are doing this alternative approach? Is the ...
0
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0answers
15 views

Antiunitary operator, momentum operator [on hold]

Assuming the time-reversal operator $T$ $T|x>=|x>$ Now I want to calculate $TpT^{-1}$ So, $TpT{-1}|x>=Tp|x>=\int\int T|x'><x'|p|p'><p'|x>=\int\int ...
0
votes
2answers
57 views

Creation and annihilation operators in Hamiltonian

If I find a Hamiltonian $H = \sum_{k} \varepsilon_k a_k^{\dagger} a_k + \sum_k V_k a_k^{\dagger} a_k$ then I was wondering: As far as I know this is many body theory and so these operators act on ...
0
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0answers
44 views

Please help with quantum mechanics [closed]

Let the Hamiltonian of two nonidentical spin 1/2 particles be where and are constants having the dimensions of energy. Find the energy levels and their degeneracies.
1
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0answers
24 views

Pure Point Spectrum implies Spanning Eigenfunctions [migrated]

If $H$ is a self-adjoint operator on a Hilbert space $\mathcal{H}$, and the spectrum of $H$ is a pure point spectrum, i.e., the spectrum consists of discrete eigenvalues (perhaps with multiplicity ...
2
votes
1answer
21 views

Ambiguity in ordering of isospin states for Clebsch-Gordan coefficients

In studying isospin for nuclear physics, I am confused a bit by an ambiguity I found. If a process that goes from $K^- + p \rightarrow \Sigma^0+ \pi^0$, I can write the isospin for the left hand side ...
0
votes
1answer
46 views

Where did I go wrong in ket vector in quantum mechanics? [closed]

Consider a system of total angular momentum j = 1. The operator jx is given by: What are the possible values when measuring jx? My attempt: The eigenvalues after calculation are -1, 0, 1. Now I ...
2
votes
1answer
81 views

Probability of energy from wavefunction

In general, given a wavefunction $\psi(x)\equiv\langle x\vert\psi\rangle$ for some system, how can one compute the probability that the system will be at a given energy level $E_n$? That is, how can ...
0
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0answers
48 views

What lives in the Hilbert Space? [duplicate]

Consider the eigenvalue equation: $$\hat{Q}\Psi = q\Psi$$ where $q$ and $\Psi$ are eigenvalues and eigenfunctions of the hermitian operator $\hat{Q}$. If the spectrum of the hermitian operator is ...
0
votes
1answer
39 views

Integral over scalar product of eigenfunction of momentum operator and harmonic oscillator one

Recently I've met following expression: $$ \tag 2 \sum_{n}f(n)\int dp~ |\langle p | n\rangle|^{2} = 2\pi \sum_{n}f(n). $$ Here $|n>$ is eigenfunction of harmonic oscillator with energy $$E_{n} = ...
0
votes
1answer
65 views

Commutation relation of position and momentum using Dirac notation

This is likely a very trivial/silly question, but in following a derivation of the position and momentum commutation relation using the dirac notation, I am having trouble justifying a certain step. ...
2
votes
2answers
73 views

What is the physical meaning of $a_{\vec{p}} \! \mid \! 0 \rangle$

$a^\dagger_{\vec{p}} \! \mid \! 0 \rangle = \mid \! p \rangle$ is interpreted as a creation of a particle with momentum $p$ from the vacuum. $a_{\vec{p}} \! \mid \! p \rangle = \mid \! 0 \rangle$ is ...
5
votes
4answers
128 views

Difference between kets $\left| -x\right\rangle$ and $-\left |x\right\rangle$?

While using dirac bra-ket notation for quantum mechanics, what's the difference if the minus sign is inside or outside the ket ? I know that $\left| -x\right\rangle$ and $- \left|x \right\rangle$ ...
0
votes
1answer
68 views

Quantum mechanics, operator commutes with Hamiltonian

My textbook said, if an operator $\hat{O}$ commutes with the Hamiltonian, then we can use the eigen vectors of the Hamiltonian as a basis of the Hilbert space, then express the operator $\hat{O}$ in ...
1
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1answer
29 views

How to identify the represented group from the basis states?

There is a 6 dimensional multiplet belonging to an irreducible representation of a unitary group of rank less than 3. How does one check if the states $|i\rangle$ belong to spin 5/2 representation of ...
1
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2answers
83 views

How would you go about evaluating $\langle \psi \mid 100 \mid \psi \rangle$? [closed]

How would you go about evaluating $\langle \psi \mid 100 \mid \psi \rangle$? I just can't seem to figure this out, and I know it isn't hard.
6
votes
2answers
134 views

Interpreting some domain issues of (potential) momentum operators

In the context of mathematical quantum mechanics, a well known no-go theorem known as Hellinger-Töplitz tells us that an unbounded, symmetric operator cannot be defined everywhere on the Hilbert space ...
1
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1answer
52 views

Proof of inability to obtain global phase

I'm curious if there's a quick proof of the inability to obtain global phase from a quantum state, since they're supposedly indistinguisable. I suppose to measure this, you would need a Hermitian ...
1
vote
4answers
267 views

Having trouble understanding some stuff about delta functions [closed]

I was going through one of the examples in Griffith's Quantum book and there was a few things in Example 3.3 that I didn't understand that I was hoping to get some clarification on. For instance, we ...
0
votes
0answers
29 views

Are complex planes (Gaussian plane) examples for a Hilbert Space?

My assumption: Yes, even though from physical application point of view it looks like different. (A complex plane is used in complex AC analysis, while Hilbert Space refers usually to QM.)
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3answers
99 views

Variational Theorem proof

I have been trying to prove variational theorem in quantum mechanics for a couple of days but I can't understand the logic behind certain steps. Here is what I have so far: \begin{equation} ...
4
votes
2answers
838 views

These two operators commute…but their eigenvectors aren't all the same. Why?

The Hamiltonian $$H = \left[ \begin{array}{cccc} a & 0 & 0 & -b \\ 0 & 0 & -b & 0\\ 0 & -b & 0 & 0\\ -b & 0 & 0 & -a \end{array} \right] $$ commutes ...
0
votes
0answers
38 views

Is it sensible to speak of the parity operator in 4 dimensional Hilbert space?

So I'm dealing with a system of two qubits, with the hamiltonian given by $$H = \left[ \begin{array}{cccc} a & 0 & -b & 0 \\ 0 & 0 & 0 & -b\\ -b & 0 & 0 & 0\\ 0 ...
4
votes
2answers
239 views

Position operator in QFT

My Professor in QFT did a move which I cannot follow: Given the state $$\hat\phi|0\rangle = \int \frac{d^3p}{(2\pi)^3 2 E_p} a^\dagger_p e^{- i p_\mu x^\mu}|0\rangle,$$ he wanted to show that this ...
0
votes
0answers
98 views

Momentum operator in Dirac formalism

Could you derive the momentum operator as follows: Since $\mathcal{T}(\Delta x)=\exp(-ip_{x} \Delta x/ \hslash)$, if we set $\Delta x=x-0$ then it follows that $\left \langle x\right | ...
7
votes
3answers
704 views

Mathematical understanding of Quantum Mechanics

Assuming that $\phi(r) = F (\psi(r))$ for some operator $F$ in Quantum Mechanics. Then, in our lecture today, we said that $$\phi(r) = \langle r|F |\psi\rangle = \int_{\mathbb{R}} \langle r |F| r' ...
2
votes
1answer
113 views

Normalization of a wavefunction that's superposition of two unknown energy eigenfunctions

Question:$$\psi(x)=A(3u_1(x)+4u_2(x))$$where $u_1(x)$ and $u_2(x)$ are energy eigenfunctions. How to normalize function $\psi(x)$? My intuitive solution: I got ...
5
votes
0answers
31 views

Motivating Irreducibility of Hilbert Space for Quantization Axioms

In the context of geometric quantization, we usually look for a map from the Poisson algebra of classical observables to the algebra of quantum observables (or rather, a sub-algebra of the classical ...
3
votes
3answers
197 views

Is $0 | \psi \rangle=0$?

For example, the spin operator for spin 1 particle is $\hat{S}_z\doteq\hbar\begin{pmatrix} 1&&\\&0&\\&&-1\end{pmatrix}$ for state ...
4
votes
2answers
76 views

Should the eigenkets be weighted in $|P\rangle = \sum\limits_{r}|\xi^r\rangle$?

Page 37 of Dirac's book The Principles of Quantum Mechanics, states The condition for the eigenstates of $\xi$ to form a complete set must thus be formulated, that any ket $|P\rangle$ can be ...
2
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2answers
80 views

Domain of simple quantum harmonic oscillator

When discussing the spectral theory of unbounded operators, one often starts with an operator defined on a densely defined subspace of your Hilbert space, and then proves that the operator is ...
1
vote
1answer
77 views

Momentum Representation vs Position Representation

We are given an operator $g$ from $\mathcal{l}^2(\mathbb{Z})$ to $\mathcal{l}^2(\mathbb{Z})$, i.e., the space of functions that are square summable over $\mathbb{Z}$ such that ...
2
votes
2answers
71 views

Is it possible that we have a physical state which is a mixture of discrete eigenstates and continuous ones?

For a system has both continuous and discrete spectrum, is it possible that a physical states is something like: ...
1
vote
1answer
77 views

Determine eigeinvalue and eigenvector of two operators R and L [closed]

Question: Let H be a Hilbert space with countable-infinite orthonormal basis ${|n>}_{n \in N}$. The two operators R and L on H are defined by their action on the basis elements \begin{align} ...
15
votes
4answers
702 views

Why do electrons in an atom occupy only the stationary states?

When we talk about the elementary problems in quantum mechanics like particle in a box, we first calculate the energy eigen-function. Then we say that the most general state is the linear combination ...
4
votes
3answers
360 views

How is a bound state defined in quantum mechanics?

How is a bound state defined in quantum mechanics for states which are not eigenstates of the Hamiltonian i.e. which do not have definite energies? Can a superposition state like ...
0
votes
1answer
125 views

Two-state Hamiltonian matrix in basis

I have a homework problem as following: Write the two-state Hamiltonian matrix in a certain basis |1>, |2> in a general form as \begin{array}{ccc} H_{11} & H_{12} \\ H_{21} & H_{22} ...
1
vote
2answers
97 views

Physical interpretation of applying a unitary operator to a state

When we apply one of the Pauli matrices $\sigma_y$ on one of its eigen-vectors $| \odot \rangle$, what does the eigen-value tell us about $| \odot \rangle$? Is this considered a measurement of $| ...
2
votes
1answer
54 views

Hilbert space decomposition into irreps

I'm currently following a course in representation theory for physicists, and I'm rather confused about irreps, and how they relate to states in Hilbert spaces. First what I think I know: If a ...
4
votes
2answers
346 views

Where to place the operator?

I believe it's standard to place the operator in between the conjugate of the wavefunction and the wavefunction itself. For instance, $$\langle p\rangle = \int_{-\infty}^{\infty}\Psi * ...
2
votes
1answer
164 views

Normalizing wavefunction

If you are trying to normalize $\psi = A\sin kx$, and you find that $|A|^2 = \frac{2}{a}$, why do you take the positive square root and not the negative? What happens if you take the negative square ...
1
vote
2answers
65 views

Could a trial wavefunction providing exact eigenenergy differ from the exact eigenfunction by a zero measure function?

Given the eigenequation of a Hamiltonian $$ H |n \rangle = E_n |n \rangle \tag{1} $$ We write it in the position representation $$ \langle x | H | n \rangle = E_n \langle x | n \rangle \tag{2} $$ ...
10
votes
4answers
205 views

What is the meaning of a state in QFT?

I guess this may be more of a mathematical than a physics question, but it comes down to physical interpretations, so I'm posting it here. In classical Quantum Mechanics, we can define a state ...
4
votes
2answers
182 views

QFT Hilbert spaces over other rings than the complex numbers $\mathbb{C}$

I would like some help evaluating a physics theory recently proposed by a physics professor at the College of Dupage. I think the theory is utterly wrong, for very simple reasons. If an amateur ...
1
vote
2answers
119 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$2$>$ ...
2
votes
1answer
67 views

How to prove that if the expectation value of $A$ in any state is real, then $A$ is Hermitian?

If the expectation value of operator $A$ in any state is real, then $A$ is Hermitian. there is an uncompleted proof: $$ \int(c_1\psi_1+c_2\psi_2)^* A (c_1\psi_1+c_2\psi_2)dx$$ ...
5
votes
3answers
101 views

Is the ground state closest to the uncertainty relation? [duplicate]

For simplicity, suppose we are only talking about discrete energy levels, ie, bound state case. The energy levels are $E_1, E_2\cdots$, and the corresponding wave functions are $\psi_1, \psi_2 ...
3
votes
0answers
70 views

Density matrix formalism and group representation

The postulates of quantum theory can be given in the density matrix formalism. States correspond to positive trace class operators with trace 1 on a Hilbert space $\mathcal{H}$. Composition is defined ...
1
vote
1answer
102 views

How does Dirac form this conjugate imaginary equation?

On page 30 of Dirac's book $$\xi|P\rangle = a|P\rangle\tag{12}$$ He then says Suppose we have a solution of (12) and we form the conjugate imaginary equation, which will read $$\langle ...