In physics, an operator is almost always either a square matrix or a linear mapping from one space of functions (often on $\mathbb{R}^N$ or $\mathbb{C}^N$) to the same or other like space of functions. Operators serve as *observables* and as *time evolution operators* in Quantum Mechanics. This tag ...

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Physical observables and hermiticity

Is it necessary for an operator to be Hermitian in order to be a physical observable or is it just sufficient that the operator obeys the eigenvalue equation? If I were to check whether an operator is ...
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About the postulates of quantum mechanics and self-adjointness

I am a freshman trying to understand the very basics of quantum mechanics but I met barriers at the beginning. What really matters is the postulates of quantum mechanics and their relationship with ...
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120 views

How does an operator transform under time reversal?

We know that a time-reversal operator $T$ can be represented as $$T=UK$$ where $U$ is some unitary operator and $K$ is the complex conjugation operator. Then under time-reversal operation, a quantum ...
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Projection operators in a direct product space

The things I'm pretty sure I understand: Let's say I have a single particle hamiltonian $H$ represented by a $2$x$2$ matrix, so it has two eigenstates $|\lambda_1\rangle$ and $|\lambda_2\rangle$. I ...
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What is the relation between renormalization and self-adjoint extension?

What is the relation between renormalization and self-adjoint extension? It seems that a renormalization scheme can be rigorously treated mathematically using the self-adjoint extension theory. Is ...
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31 views

Addition of angular momenta, coefficient in the |10> state

In Griffith's text, they apply the lowering operator on the |$11\rangle$ state to get the |$10\rangle$ state. They show this result in two forms on pg. 185: $S_{-}\left(\uparrow\uparrow\right) = ...
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String Excited States / Spectrum

I am working through David Tong's lecture notes on string theory. The bit I am stuck on is page 41/42 in Chapter 2. The PDF file is here. In 2.3.2 "The First Excited States" on page 41/42, he says: ...
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Eigenvalues of Angular Momentum in Quantum Mechanics

The eigenvalue equation of the $L^2$ operator is given by $$L^2f_l^m = \hbar ^2l(l+1)f_l^m$$ Side: So a determinate state for some observable $Q$ is a state where every measurement of $Q$ returns ...
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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 ...
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Finding the expectation value of the annihilation operator with respect to a given state

Using dirac notation we were given a state vector $$|\Psi(t=0)\rangle = A\sum\limits_{q=0}^Q \frac{1}{(q+i)} |\phi_q\rangle$$ Where $\phi$ is part of a complete orthonormal set. I found the ...
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Question about derivation of the Heisenberg Uncertainty Principle?

I am looking at the derivation presented here. The first thing I am unsure about is where the form of $\psi_0=Ae^{\frac{-m\omega x^2}{2\hbar}}$ came from. Also, is this form for all $\psi$, or just ...
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Does the angular momentum vector operator $\hat{\vec{J}}$ have no eigenstates?

Angular momentum projection operators $\hat J_z$ and $\hat J_y$ don't commute, as don't the other combinations of different projections. But this means that there's no such state in which the whole ...
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81 views

Creation and annihilation operators

In our lecture today, we introduced two kinds of creation and annihilation operators. I want to restrict myself to the antisymmetric case: The first operator $a_k^{\dagger}$ creates a state ...
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77 views

Uncertainty relation for non-simultaneous observation

Heisenberg's uncertainty relation in the Robertson-Schroedinger formulation is written as, $$\sigma_A^2 \sigma_B^2 \geq |\frac{1}{2} \langle\{\hat A, \hat B\}\rangle -\langle \hat A\rangle\langle ...
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106 views

Second quantization, creation and annihilation operators

I found two notions of states for second quantization. One representation uses occupation numbers here, for example Another one creates the n+1 th particle in a collection of n existent states. see ...
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Why does the measurement of some observable $A$, the measured value is always an eigenvalue of the operator?

Explain why when we make a measurement of some observable $A$ in QM, the measured value is always an eigenvalue of the operator $A$.
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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 ...
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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. ...
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Expectation value expression Quantum Mechanics

Whilst working on a project I kept stumbeling across two different expressions for the standard deviation $\Delta{X}^2 = <(X - <X>)^2 >$ and the other $\Delta{X}^2 = <X^2> - ...
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Confusion with rotation operator definition in Shankar

In Shankar quantum mechanics on page 306-307 it has the following: 12.2. Rotations in Two Dimensions Classically, the effect of a rotation $\phi_0\mathbf{k}$, i.e., by an angle $\phi_0$ about ...
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Unitary change of X Basis: Shankar Exercise 7.4.9 [closed]

I'm currently working through Shankar's Quantum Mechanics and am stuck on one of his exercises. In Exercise 7.4.9 Shankar would like us to show that: if $\mid x \rangle$ is changed to $\mid ...
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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 ...
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Apparent spacetime dependence of creation and annihilation operators

I'm currently going through An Introduction to Quantum Field Theory by Hartmut Wittig I've stumbled upon. Having trouble with equation (2.29), I'm asking the question: Do creation and annihilation ...
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59 views

Time derivative of $\hat p$ under time varying Hamiltonian

How does one show that $$ \frac {d\hat p}{dt} = \frac 1{i\hbar}[\hat p, \hat H]$$ is valid even when Hamiltonian is time dependent explicitly? I can see that this is true when $\hat H$ is time ...
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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 ...
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54 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 ...
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How to compute the normal ordered angular momentum of a Klein-Gordon real scalar in terms of ladder operators?

I'm trying to compute the angular momentum $$Q_i=-2\epsilon_{ijk}\int{d^3x}\,x^kT^{0j}\tag{1}$$ where ...
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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 ...
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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 ...
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67 views

Finding the spectrum of a curious hamiltonian

I wish to analyse the following hamiltonian, i.e. find its eigenvalues and eigenstates. $$H = \frac{1}{2}\epsilon(\sigma _z \otimes \mathbb{1} + 1\otimes \sigma _z) - \Delta (\sigma _x \otimes \sigma ...
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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 | ...
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Where does the partial derivative come from in Sakurai's derivation of the momentum operator?

How is the momentum operator derived in Dirac formalism? I am reading Quantum Mechanics by Sakurai and he gives the following derivation. But I don't understand how he goes from the third equation to ...
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Defining creation and annihilation operators

Creation and annihilation operators can be defined in several different ways, some more general than others. We usually choose to denote by $a$ the annihilation operator and by $a^\dagger$ the ...
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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' ...
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Constructing matrix for spin in Stern-Gerlach experiment for arbitrary angle

This is a conceptual question about a problem in Sakurai. I understand how to solve the problem, but there's something about it that irks me, and it feels like I'm missing something. In the problem, ...
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Eigenvalues being physical observables

I think I'm comfortable with the PDE solutions to the Schrodinger equation. But as soon as we start putting these values in a matrix (in dirac notation), I lose my understanding and everything ...
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209 views

Grassmann number representation for fermions

How one can simultaneously represent fermionic operators and its corresponding Grassmann variables, so that all the anticommutation relations between them and also states would take place? $$ ...
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112 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 ...
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Split property for type III algebras entails practical separability

I am reading Halvorson's thesis (http://philsci-archive.pitt.edu/346/1/main-new.pdf), however I don't understand a proof at p.50 where he tries to explain why the split property allows a local agent ...
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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 ...
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Does the commutator of anything with itself not vanish?

In a quantum mechanics exam one question was to write the commutator of a couple of operators. Everybody got points taken away since they did not write $[Q_i, Q_i] = 0$ for all the operators $Q_i$ in ...
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Kraus operator rank

All quantum operations $\mathcal{E}$ on a system of Hilbert space dimension $\mathcal{d}$ can be generated by an operator-sum representation containing at most $\mathcal{d^2}$ elements. Extending ...
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How to derive $[x_i, F(\vec p)] = i \hbar \frac {\partial F(\vec p)}{\partial p_i}$

Wikipedia indicates that the following relation is "easily shown": $[x_i, F(\vec p)] = i \hbar \frac {\partial F(\vec p)}{\partial p_i}$, however I'm having some trouble showing it. I think I'm just ...
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334 views

Intuition on positive-operator valued measures (POVM)

I'm having a little trouble understanding what positive-operator valued measure (POVM) are- in particular why/how they are non-negative. For instance, if they just represent measurements, what about ...
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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} ...
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Translational operator on potential

In https://wiki.oulu.fi/download/attachments/14553161/lattice.pdf I have a problem with the translational operator: The second line under the first figure says $$\tau^\dagger(a)V(x)\tau(a)=V(x+a).$$ ...
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An Operator Identity relating to Trace [duplicate]

Suppose that $\hat H$ is an operator (typically a Hamiltonian) and $\beta$ is a positive parameter (typically $\beta=1/k_BT$). Show that $$ \mathbf{Tr}\Big(e^{-\beta\hat H}\Big) \geq ...
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Does there exist a state for which $\Delta\sigma_x^2=\Delta\sigma_y^2=0$? If not, how does one prove it?

I just realized that the uncertainty principle says that $$\Delta\sigma_x^2 \Delta\sigma_y^2 \ge \left(\overline{\hat\sigma_z}\right)^2,$$ where ...
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Classical Hamiltonian involving product of factors whose quantum analogues don't commute

Dirac remarked in his quantum mechanics book: One can usually assume that the Hamiltonian is the same function of the canonical coordinates and momenta in the quantum theory as in the ...
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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 * ...