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46 views

What is the implication of overlap between eigenstates of two operators in Quantum Mechanics?

For instance, what does it mean that a certain position eigenstate is also an energy eigenstate? I understand that measurable (Observables) in Quantum mechanics are the operators. Their eigenvalues ...
2
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1answer
38 views

Commutation relations in Gupta-Bleuler quantization

Quantization of the free electro-magnetic field has essential differences in comparison to quantization of say scalar or massive vector fields. In fact there are different approches to it. One of ...
-1
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1answer
63 views

Proof for $\langle i[A,B]\rangle$ [closed]

I have to prove the following equation: $$ \langle i[A,B]\rangle = 2\mathfrak{Im}\left[\int dV(\overline{B\psi)}(A\psi)\right]\,,$$ where A,B are hermitian operators. Here is my calculation, but I don'...
0
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0answers
21 views

Source for mathematical methods [duplicate]

I am just curious about any question and example sources for linear vector spaces, bra-ket notation, operators, commutators and hilbert spaces.
1
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1answer
66 views

Does $[L_z,H] = 0$ imply the state is also an Eigenstate of $H$ is also an eigenstate of $L_z$?

Given that the Hamiltonian $\mathcal{H}$ is rotationally invariant then we know $[L_z,\mathcal{H}] = 0$. Does that imply that an eigenstate of H is also an eigenstate of $\mathcal{H}$? More ...
1
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1answer
53 views

Does total $\hat{S}^2$ always commute with total $\hat{S}_z$ even for interacting spins?

I was given the following operator $\hat{f}$ describing the interaction of two spin-$\frac12$ particles: $$\hat{f}=a+b{\hat{\bf S}_1}\cdot{\hat{\bf S}_2}.$$ I was told that I can prove that $\hat{f}$...
0
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1answer
48 views

The charge given by a commutator

I saw in the text that $[Q,X]=cX$ and says the operator $X$ has charge $c$ under the generator $Q$. I tried to understand why the coefficient $c$ means the charge. So I used this relation to get the ...
0
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1answer
24 views

Symmetry in Fock-space 2-body interaction

The simplest two body interaction term for fermions is $$H = \sum_{ijkl} U_{ijkl} a_i^\dagger a_j^\dagger a_k a_l$$ and I'm trying to determine the symmetries on $U$. Unfortunately I keep getting ...
2
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2answers
83 views

Angular momentum coupling

I read about angular momentum coupling on wikipedia and there are a few things i dont understand. What does this mean "spin and orbital angular momentum of a single object belong to different Hilbert ...
0
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0answers
66 views

What is the QFT state with two distinguishable fermions present?

I want to describe a system with two non-interacting and definitely different fermions, say an electron neutrino, $\nu_e$, and an electron, $e^-$. The state describing a single electron is given ...
1
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1answer
164 views

“Commuting observables share common eigenstates”

I am struggling to find a precise definition of this line from my quantum mechanics textbook: If $[A,B] = 0$, then the operators commute, and "commuting operators share common eigenstates". This ...
2
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3answers
133 views

Operators commutation and relation between eigenvalues

If $H$ and $L_i$ are commuting ( $[H, L_i] = 0$ ) could we deduce that the eigenvalues of $H$ depend/ do not depend on $m$ and $\ell$ ( eigenvalue of $L_z, L^2$ )? I don't think so since it does not ...
2
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1answer
72 views

Joint Spectral Measure theorem

I want to gain an intuition to understand the joint spectral measure theorem. In the case that operators involved in this theorem have purely discrete spectrum, the theorem should be reduced to the ...
1
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1answer
144 views

Ladder operators vs creation/annihilation operators

I am trying to figure out the difference between the ladder operators (for harmonic oscillator) $a^\dagger$, $a$ and the creating/annihilation operators $c^\dagger$, $c$. Are they the same? I have ...
2
votes
2answers
186 views

What does it mean for 2 observables to be compatible?

If I have 2 observable operators $A$ and $B$, if $A$ and $B$ commute: $[A, B] = 0$, then they must necessarily form a complete set of commuting observables (CSCO). Essentially, if 2 observables are ...
6
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2answers
377 views

Commutator expectation value in quantum mechanics

Suppose $A$ and $B$ are operators, $A$ is Hermitian, $B$ anti-hermitian, and their commutator is the identity, i.e. $$[A, B] = I \, .$$ Denoting the eigenvectors of $A$ as $\lvert a \rangle$, so that $...
0
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1answer
72 views

If $\psi$ acting on $a_+$ and $a_-$ operator just moves up and down the ladder, why is $[a_-, a_+] = 1$ and not 0? [duplicate]

If $\psi$ is acted upon by both the operators one by one, it should return the same wave function. Thus order in which you increase or decrease the energy shouldn't matter. Then why is it so that the ...
1
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1answer
187 views

No sense in the expression $\hat{x}| 1\rangle=\sqrt{\frac{2}{a}}\int_{-\frac{a}{2}}^{\frac{a}{2}}x\cos\left(\frac{\pi}{a}x\right)dx=0$

I am considering a particle of mass m in a symmetric infinite square well of width a in the fundamental state. $$V(x)= \begin{cases} 0 & \mbox{$|x|<\frac{a}{2}$} \\ \infty & \mbox{...
1
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1answer
48 views

Result of the measurement of operators $A$ and $B$ on same state $|\psi\rangle$ if $[A,B]=0$

Consider a 3-dimensional Hilbert space spanned by the normalized eigenstates $|1\rangle,|2\rangle,|3\rangle$ of an operator $A$. Consider a normalized superposition, $|\psi\rangle=c_1|1\rangle+c_2|2\...
0
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2answers
198 views

If operators do not commute they cannot have a complete set of eigenfunctions [closed]

$\def\ket#1{|#1\rangle}$ I am trying to show a simple proof of, If operators do not commute they cannot have a complete set of eigenfunctions. start Let $[\hat P, \hat Q]=0$, or $\hat P \text{ ...
2
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1answer
121 views

Non-Fock representation of quantum field theory

I cannot find reference, but I read that in curved spacetime, there exists representation that is not Fock satisfying CCR and unitarily inequivalent to a Fock representation. In usual understanding ...
-1
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1answer
459 views

Simultaneous measurement of two observables

In quantum physics the configuration of a particle is fully defined by it's wave function. When a measurement of a particular observable ( eg. position, angular momentum etc.) is made on the particle ,...
8
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2answers
664 views

How to know if a set of commuting observables is complete?

We define a complete set of commuting observables as a set of observables $\{A_1,\ldots, A_n\}$ such that: $\left[A_i, A_j\right]=0$, for every $1\leq i,~j \leq n$; If $a_1,\ldots, a_n$ are ...
-2
votes
1answer
111 views

Common basis of Hamiltonian and another operator [closed]

I have a Hamiltonian of $N\times N$ matrix, when I diagonalize it , I observe that the $N$ different eigenstates of the hamiltonian are also eigenstates of another operator $\hat O$ where the ...
5
votes
3answers
533 views

Stone-von Neumann theorem

According to Stone-von Neumann theorem, any two canonically conjugate self adjoint operators following the relation: $$[\hat{q},\hat{p}]=i\hbar$$ cannot be both bounded. I am confused about how we ...
3
votes
1answer
305 views

Is there any plausible reason for the existence of conjugate variables in quantum mechanics?

(I updated the question, the new last passage is the important one) If I assume the position of a particle to be represented by an operator $\hat{x}$, and the time evolution to be carried out by the ...
0
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2answers
831 views

Hermitian operator followed by another hermitian operator – is it also hermitian?

Consider the two hermitian operators $\hat A$ and $\hat H$: I can prove that the operator $[\hat A,\hat H]$ is non-hermitian as follows: $$\begin{align} \int\phi^*[\hat A,\hat H]\psi\,dx&=\int\...
3
votes
4answers
89 views

Quantum State Representation with Commuting Operators

Let $[A,B]=0$. Then, we can find a set of eigenvectors $\{|a_n,b_n\rangle\}$ common to both $A$ and $B$. According to this, and my own understanding, it makes sense to write an arbitrary quantum state ...
7
votes
1answer
351 views

Fock space with mixed anti-commutation/commutation relations?

Let's say we have two modes, with the following labeling of occupation number states: $ \lvert \Psi \rangle = \begin{pmatrix} 0,0 \\ 0,1 \\ 1,0 \\ 1,1 \end{pmatrix} $ An example of (what I assume to ...
1
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1answer
82 views

Heisenberg Uncertainty relation

The derivation of the uncertainty principle says that for any 2 observables $A$ and $B $, we act $$ A+i\lambda B $$ on a normalized state and demand the norm of the new state be greater than or ...
0
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1answer
200 views

If the mean of the commutator is zero, then what we can say about the commutator itself?

Suppose we have \begin{equation} \langle[H,N]\rangle=0 \tag{1} \end{equation} where both $H$ and $N$ are hermitian. Under which assumption I can claim that then $$[H,N]=0~?\tag{2}$$
0
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2answers
335 views

Simultaneous eigenkets of angular momentum operators in QM

Is it the case that if you have simultaneous eigenkets say $|l m \rangle$ then we have both $$\sum_{l} |l m \rangle \langle l m \rangle = I ~~\text{ and }~~ \sum_{m} |l m \rangle \langle l m \rangle = ...
1
vote
1answer
175 views

Commutator with subspaces belonging to the same eigenvalue

I have two operators, which commute with each other: $[A,B]=0$. The operator $A$ is Hermitian and has a degenerate spectrum. Does it follow from $[A,B]=0$ that the operator $B$ commutes with every ...
1
vote
1answer
177 views

Can someone help me with a proof of the expectation value of the dispersion of an observable A only using Dirac Notation?

Unfortunately I learned Quantum Mechanics from Griffiths and I struggle greatly with Dirac notation. I'm working my way through Sakurai learning it and tried to do a very trivial proof using only ...
2
votes
1answer
113 views

Restriction of an operator

I am reading Cohen-Tannoudji's Quantum Mechanics Vol. 1. In problem 11 from Chapter II there are two given operators defined in a space generated by $\{\left|u_1\right>,\left|u_2\right>,\left|...
2
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0answers
388 views

Non-commutativity of operators, simultaneous eigenstates and degeneracies over finite dimensional Hilbert space

Suppose, I have two operators $\hat{A}$ and $\hat{B}$ defined over a finite dimensional Hilbert space $H$. Also assume that both the operators have their own eigenstates that span $H$(existence of ...
0
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1answer
379 views

Verification of proof of complete set of commuting operators

Hi I am interested in the validity of the following proof. I am interested in the validity of this particular proof as I am aware of how to prove this result in a different way. Theorem: If two ...
3
votes
2answers
853 views

Why do the ladder operators in harmonic oscillators work?

The Hamiltonian can be diagonalized by transforming $x$ and $p$ to $a$ and $a^\dagger$. I understand how one proceeds from there to find the spectrum of $a^\dagger a$, the ground state $|0\rangle$ and ...
0
votes
1answer
96 views

Question about a formula in the book by Green, Schwarz, Witten

In chapter 7 of superstring theory, it is written $$ g\langle0;k_1|\zeta\cdot\alpha_1V_0(k_2)\zeta_3\cdot\alpha_{-1}|0;k_3\rangle=g\langle0;k_1|\zeta\cdot\alpha_1e^{k_2\cdot\alpha_{-1}} e^{-k_2\cdot\...
1
vote
1answer
1k views

Tensor products of Hilberts spaces: definition of outer products and commutators

Suppose one has two single-particle Hilbert spaces $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ and consider the tensor product of these such that $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ is a two-particle ...
0
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0answers
360 views

Physical significance of non-commutativity of ladder operators in Quantum Harmonic Oscillator

If we apply the raising (creation) operator to $Ψ_n(x)$ and the apply to it the lowering (annihilation) operator, we get $Ψ_n(x)$ times a constant. Does it physically say something? Can we get any ...
3
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1answer
760 views

Commutation relations in quantum mechanics

As we know, simple harmonic oscillator can be solved only by commutation relations between creation and annihilation operators, and the Hamiltonian expression. The spin energy is either solved only ...
4
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3answers
1k views

Eigenstates of Ladder Operators

According ot Griffith's Intro to Quantum Mechanics (page 147), if some function $f$ is an eigenfunction of $L^{2}$, then $L_{-}f$ is also an eigenfunction of $L^{2}$. Is $f$ also an eigenfunction ...
1
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1answer
3k views

Ladder operators - commutation relations and their properties

At the beginning of Fetter, Walecka "Many body quantum mechanics" there is a statement, that every property of creation and annihilation operators comes from their commutation relation (I'm ...
3
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0answers
252 views

commutation relations in terms of eigenstates scalar product

This question has caught my attention because I was unaware of the fact that the position-momentum canonical commutation relations could be derived out of the only assumption for $\langle x | p\rangle$...
2
votes
1answer
233 views

Second Quantization: The Identity Operator does not Commute?

Let me take the simplest possible example. Consider the fermonic Fock-space $\Lambda^*(\mathbb{C}^n)$ built out of a finite-dimensional, oriented single-particle Hilbert space $\mathbb{C}^n$ with ...
1
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1answer
851 views

Expectation value of an operator and commutator relation

I have a quantum operator $A.$ It's expectation value is constant respect to time. I mean $$\langle \psi(t)|A|\psi(t)\rangle$$ is a constant values. If I know $|\psi(t)\rangle$ is not an eigenstate ...
2
votes
2answers
826 views

Commutator of position and momentum

I'm reading Sakurai's Quantum Mechanics. One of the problem in the book asks to use the relation $$ \langle{x}|p\rangle=\frac{1}{\sqrt{2\pi\hbar}}e^{\frac{ipx}{\hbar}} $$ to evaluate $\langle{x}|[X,P]|...
4
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2answers
353 views

$\hat{L}_{x}$ and $\hat{L}_{y}$ do not commute… or do they?

So $\hat{L}_{x}$ and $\hat{L}_{y}$ do not commute: $$ [ \hat{L}_{x}, \hat{L}_{y}] = i\hbar \hat{L}_{z}$$ But, what if we perform this operation on a state such that: $$\hat{L}_{z} \phi_{l, m_{l}} =...
1
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0answers
648 views

Simultaneous eigenket

J. J. Sakurai states in his "Modern Quantum Mechanics", this fact as a theorem ($\pi$ is the parity operator): Suppose $$[H,\pi]=0$$ and $| n>$ is a nondegenerate eigenket of $H$ with ...