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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Density matrix formalism

The density matrix $\hat{\rho}$ is often introduced in textbooks as a mathematical convenience that allows us to describe quantum systems in which there is some level of missing information. ...
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Applying an operator to a function vs. a (ket) vector

I have a question regarding the effect of quantum mechanical operators. The definition that I'm familiar with says that an operator $A$ acts on a vector from a Hilbert space, $|\psi\rangle$, and the ...
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28 views

Expectation value and Dispersion of an Operator

Suppose we have an operator $Q$ with eigenvalue $q$. Expectation value is $\langle Q \rangle$ and dispersion $D(Q) = \sqrt{\langle \left( Q - \langle Q \rangle \right)^2 \rangle} $. I want to find ...
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79 views

Inner product of position and momentum eigenkets

Let's define $\hat{q},\ \hat{p}$ the positon and momentum quantum operators, $\hat{a}$ the annihilation operator and $\hat{a}_1,\ \hat{a}_2$ with its real and imaginary part, such that $$ \hat{a} = ...
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Experimental proof of the principle of superposition in QM [duplicate]

I have read that we need all operators in QM to be linear to confirm the principle of superposition which is experimentally well proven. I wonder how such an experiment could be made?
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Nuclear Physics: Eigenvalues of nucleus angular moment

In the shell model of nuclei, when we talk about collective motions, we describe any nucleus deformation, expanding its radius on spherical harmonics base like this $R(\theta,\phi) = ...
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46 views

Transition Between Position and Momentum Basis

I'm having some trouble following pages 55-56 of Sakurai's Modern Quantum Mechanics. We're trying to transfer from position space into momentum space. Here's a quote: Let us now establish the ...
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3answers
90 views

Can one construct a new operator in terms of the powers of another operator?

Suppose we have a quantum state, well described by its time-independent wave function Psi. And we have a well-defined Hermitian (self-adjoint) operator $A$. We successfully evaluate the expectation ...
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75 views

Trace operation on dynamic equation: physical meaning?

Suppose we have Heisenberg equation of motion for some observable $A$, $$ i\hbar\frac{dA}{dt}= -[H,A] $$ since the trace of any finite dimensional commutator structure vanish(not something like ...
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887 views

Is the Uncertainty Principle valid for information about the past?

My layman understanding of the Uncertainty Principle is that you can't determine the both the position and momentum of a particle at the same point in time, because measuring one variable changes the ...
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How can $J_1^2, J_2^2, J_{1z}, J_{2z}$ commute mutually?

I'm reading through J. J. Sakurai's Modern Quantum Mechanics book and currently looking at the "Angular-momentum addition" part. Here, it says you have two options and that one option is to ...
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Self-adjoint extensions with 'teletransporting' boundary conditions

When choosing a self-adjoint extension of a Hamiltonian, in general one can obtain domains in which (i) the probabilities teleport* between points on the boundary and (ii) boundary conditions ...
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Perturbation of an operator - Meaning of matrix element [closed]

Let be $B$ an operator and $\left|\Psi\right>$, $\left|\Phi\right>$ two states (not necessarily equals). What is the meaning of a matrix element $\left<\Psi\right| B ...
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319 views

Annihilation and Creation operators not hermitian

The annihilation and creation operators are given below $$ \hat a|n\rangle=\sqrt{n}|n-1\rangle\qquad\text{and}\qquad\hat a^\dagger|n\rangle=\sqrt{n+1}|n+1\rangle $$ Now I know the operator a^ is not ...
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Is there is a specific original source where the “quantum operator ordering issue” is stated?

During my research, when the quantum operator ordering ambiguity is mentioned is deemed usually in the likes of "the well-known problem of ordering in quantum mechanics". However, could anybody point ...
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79 views

Expectation value of an operator

Suppose we have: $$ \hat{Q}|\psi_1\rangle=q_1|\psi_1\rangle \\ \hat{Q}|\psi_2\rangle=q_2|\psi_2\rangle $$ with $q_1 \neq q_2$. Then consider the state: $$ ...
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Representations in quantum mechanics [closed]

This might be a very simple question. I just want someone to point me the right direction to understand things like this: $$ \langle x|x'\rangle=\delta(x-x') \\ \psi(x)=\langle x|\psi\rangle \\ ...
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1answer
32 views

Jump Method and the Lindblad Equation

I am studying the time evolution of a density matrix using the Lindblad equation. My initial density matrix is $\rho(0)=|\alpha\rangle\langle\alpha|$, where $|\alpha\rangle$ is a coherent state. Then ...
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76 views

Does the average momentum vanish for an eigenstate of the simple harmonic oscillator?

Suppose we have a simple harmonic oscillator, let's consider the ground state, $|0\rangle$ and the first excited state $|1\rangle$. $\langle 0|\hat p|0 \rangle$ is zero right? Since the particle can ...
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1answer
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Time evolution of states - Is total energy constant or not?

Suppose the state of the particle is given as follows: $$ |\psi_{(t)}\rangle = \frac{1}{\sqrt2} \left( e^{-\frac{i\omega t}{2}} |0\rangle + e^{-\frac{3i\omega t}{2}} |1\rangle \right) $$ Where the ...
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1answer
51 views

Calculations with operators - Proof: Equation of Operators [duplicate]

I have a problem in Quantum mechanics 1 with Operators. I have to prove the following equation. I tried it for about 4 hours without any result: Condition: $[[\hat A,\hat B],\hat A]=[[\hat A,\hat ...
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2answers
148 views

Angular momentum - maximum and minimum values for $m_{\ell}$

I want to work out the maximum and minimum values for $m_{\ell}$. I know that $\lambda \geq m_{\ell}$, therefore $m_{\ell}$ is bounded. In the lectures notes there is the following assumption: $$ ...
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1answer
35 views

Ladder operator on momentum basis

Since in Quantum mechanics momentum operator can be written in terms of ladder operators $$\widehat{p}=-i\sqrt\frac{{\hbar m \omega}}{2}(\widehat{a}-\widehat{a}^\dagger)$$ these operators operate on ...
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185 views

What is $\hat{p}|x\rangle$?

Trying to solve a QM harmonic oscillator problem I found out I need to calculate $\hat{p}|x\rangle$, where $\hat{p}$ is the momentum operator and $|x\rangle$ is an eigenstate of the position operator, ...
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2answers
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Derivation of a quantum dynamical map on open quantum system

Let us consider a initial total quantum system as $\rho(0) = \rho_S(0)\otimes\rho_B$ where $\rho_S(0)$ is initial open system and $\rho_B$ is density matrix for environment. We can use partial trace ...
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Examples of Weyl transforms of nontrivial operators

I've been able to find examples of Weyl transforms of operators like $\hat{x}$,$\hat{p}$, and $\hat{1}$, but not anything more complicated. Are there derivations of the Weyl transforms of more ...
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Query on an operator acting on a function

I have a naive question about an operator acting on a well-behaved function. Let us say, we are talking about space translation operator acting on a function $\psi(x)$: ...
4
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1answer
130 views

How are matrices used to represent quantities, and what is the meaning of a matrix?

So I'm reading this text on Quantum Mechanics, and it goes through a few chapters that I understand fairly well including probability. But then it says that all quantities, like position and energy ...
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60 views

Commutation Relations for Creation & Annihilation Opertors of Two Different Scalar Fields

Let us consider two different scalar fields $\phi$ and $\chi$. The commutation relations for the creation and annihilation operators of the scalar field $\phi$ are given by $$ [a(\textbf{k}), ...
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How to make a base tranformation for a linear operator in QM? [closed]

I have 2 bases A and B with the following kets: Base A: $|a_1\rangle$ and $|a_2\rangle$ Base B: $|b_1\rangle = \frac{1}{\sqrt2} \cdot(|a_1\rangle + i\cdot|a_2\rangle)$ $|b_2\rangle = ...
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255 views

Evaluation of expectation values

I will denote operators with hats. Suppose we got an operator of the form $i[\hat p, \tan^{-1}(e^{\hat x})]$ and we want to calculate the amplitude for a transition from a state $|p_i\rangle$ to the ...
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Galilean Transform

I tried to solve a problem using two different ways and I had some trouble, the problem is: We define a symmetry transform of the expected value of $\vec{P}$ like this: $$\langle \psi|\vec{P}|\psi ...
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Are there two different ways to express the position operator $x$ in terms of the creation and annihilation operator?

As we known, to express the position operator $x$ in terms of the creation and annihilation operator $a^{+}$ and $a$, one way is: $$x= \sqrt{\frac{\hbar}{2\mu\omega}}(a^++a);$$ $$p= ...
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1answer
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How does $p_x$ commute with $p_y$, i.e. $[p_x,p_y]=0$? [closed]

I know it's a simple and basic question but would someone show me how to evaluate $[\hat{p}_x,\hat{p}_y]$?
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1answer
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Lippmann-Schwinger solution

What's wrong with this general solution of the Lippmann-Schwinger equation: $$ |\psi_k \rangle=|\phi_k \rangle+G_k V|\psi_k \rangle $$ Taking the inner product with $\langle\phi_{k'}|$ \begin{align} ...
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Why is momentum quantized in a 1D box even though the operator doesn't give eigenstates?

We don't get eigenstates of momentum when we operate momentum operator in the wave function of particle in a 1D box problem yet we say momentum is quantized in this situation. Why is it so?
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Trouble with proof about operators in QM

I would like to complete the following exercise: Prove that if the operators $P_i$ satisfy $P_i^{\dagger}$ = $P_i$ and $P_i^2$ = $P_i$, then $P_iP_j=0$ for all $i\neq j$. From $P_i^2 = P_i$ I ...
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1answer
72 views

Angular Momentum in Quantum mechanics

In Gasiorowicz's Quantum Physics, we determined the relation: $$L_z | l,m\rangle= \hbar m | l,m \rangle$$ I would like to determine: $\langle l,m_1 | L_x | l,m_2 \rangle $ I thought about expressing ...
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1answer
54 views

Probability of measuring two observables in a mixed state

Lets say i have density Matrix on the usual base $$ \rho = \left( \begin{array}{cccc} \frac{3}{14} & \frac{3}{14} & 0 & 0 \\ \frac{3}{14} & \frac{3}{14} & 0 & 0 \\ 0 & ...
4
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262 views

Complex conjugate of momentum operator

Consider momentum operator representation in position space. $$\hat{p}=-i\frac{\partial}{\partial x} \,\ \text{and its eigen functions are } e^{ipx} \,\text{and} \,\ e^{-ipx}.$$ ...
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1answer
54 views

Is there any non-hermitian operator on Hilbert Space with all real eigenvalues?

The property of hermitian is the sufficient condition for eigenvalue being real. Is there any non-hermitian operator on Hilbert Space with all real eigenvalues? If there exist, then can all ...
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1answer
84 views

The correspondence between Grassmann number and 4-spinor

In canonical quantization, we view the Dirac field $\psi$ as a $4\times1$ matrix of complex number. While in path integral quantization, we view the Dirac field $\psi$ as a Grassmann number. For two ...
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125 views

Eigenstate of field operator in QFT

Why don't people discuss the eigenstate of the field operator? For example, the real scalar field the field operator is Hermitian, so its eigenstate is an observable quantity.
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Ordering of differential operators

If we write something like: $\partial_a X_{\mu} \partial^a X^{\mu}$ Does that mean the first derivative is only applied to the first X? ($\partial_a X_{\mu})( \partial^a X^{\mu}$) Or is the ...
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Quantum Mechanics - Observable

If $O$ represents an operator corresponding to an observable why does the following equality hold? $$\langle f(x)\, |\, O g(x)\rangle = \langle g(x) \,|\, O f(x) \rangle$$ It is used on the last ...
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Is obtaining the coordinate representation of momentum operator from commutator more fundamental than generator of translation

Related post: What is the most general expression for the coordinate representation of momentum operator? There are two methods of obtaining the coordinate representation of momentum in quantum ...
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1answer
138 views

Time-Energy Uncertainty Principle and Operators

In most of examples, I notice that uncertainty principle for time & energy is given between mass & lifetime. The UP for time and energy is $$ \Delta t\,\Delta E\geq\frac h{4π} $$ where $$Δt ...
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1answer
169 views

Diagonalization of Hamiltonian

Typically, one way of understanding the physics of an interacting quantum system is by diagonalizing the Hamiltonian. In principle, can we always diagonalize a Hamiltonian, such that it is expressed ...
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1answer
158 views

Normalized Projection Operator

What is meant by normalized projection operator? What is its physical meaning in quantum mechanics? I am pretty confused regarding the physical interpretation of both projection operator and ...
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102 views

Projection operators and their subspaces (of Hilbert space)

I've been watching Susskind's lectures on Quantum Entanglement, and something he said regarding (non-)commuting projection operators confused me. Consider two subspaces {$|a>$} and {$|b>$} of ...