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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Normal ordering

If I understood correctly there are two terms called normal ordering: $:c c^\dagger: = c^\dagger c \hspace{.5cm}$so shifting all creation operators to the left and all annihilation operators to the ...
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How to write QM operator if I know all of it's eigenfunctions?

Suppose I have selected enough orthogonal functions in representation of operator A and I want to derive operator B which has ...
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What does QM observable operator actually DO?

I know, that each observable is represented by some linear operator, while it's eigenvalues represent probability amplitudes of possible observable values. But what if I ACT with this operator on ...
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How to write “postselection” operator?

Suppose, I wish to know an operator, which eigenvalue is 1 if state is exactly F and 0 ...
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Is mass an observable in Quantum Mechanics?

One of the postulates of QM mechanics is that any observable is described mathematically by a hermitian linear operator. I suppose that an observable means a quantity that can be measured. The mass ...
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Symmetry and Algebra

I'm trying to get a more concrete idea how symmetry is understood in quantum theories, as broad as possible. Consider a infinitesimal transformation of states in quantum physics of the form: $$ ...
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479 views

Differences between probability density and expectation value of position

The expression $\int | \Psi\left(x\right)|^2dx$ gives the probability of finding a particle at a given position. If wave function gives the probabilities of positions, why do we calculate ...
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Eigenstates of a shifted harmonic oscillator

Let's say I have a quantum harmonic oscillator $H = \omega a^\dagger a$, where $a^\dagger$ is the raising operator and $a$ is the lowering operator and $H |n\rangle = \omega n |n\rangle$. Now assume ...
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Unitary transformation behind gauge transformation

It is very well-known that for bosonic operators a Gauge transformation can always be associated with it $$a\rightarrow e^{i\phi}a.$$ Obviously this is a Unitary transformation. Something like ...
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Does the order of variables matter for a quantum Lagrangian in the path integral formula for quantum mechanics? [duplicate]

For a single particle or field, I can't see how the path-integral formulation depends on the order of terms in the Lagrangian. It seems that you integrate the classical Lagrangian to get the action on ...
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Differentiation operator with respect to observable acting as a function of the observable?

In his Principles of Quantum Mechanics Dirac writes: $$\int \langle \phi \frac{d}{dq}|q'\rangle dq' \psi(q')=\int \phi(q') dq' \frac{d\psi(q')}{dq'}.$$ To me it is rather strange, and it seems as if ...
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Commutator with Pauli spin matrices and the momentum operator

How is $\left[\vec\sigma \cdot \vec p, \vec \sigma \right]$ proportional to $\vec \sigma\times \vec p$, where $\sigma$ are the Pauli spin matrices and $p$ is the momentum operator?
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Can we correctly define momentum operator only by means of position operator and their commutation relation?

In "J.M. Ziman. Electrons and Phonons: The Theory of Transport Phenomena in Solids" the author formally introduces the position (displacement) operator and then defines the momentum operator with the ...
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Observables in Quantum Mechanics

Studying on own quantum mechanics I came across: Preceeding text: A basic postulate of quantum mechanics tells us how to set up the operator corresponding to a given observable. Observables, ...
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Is quantum field operator $\psi$ same as quantum field $\psi$?

So in QFT, quantum field operator $\psi$ is there. $\psi$ seems to take the role of wavefunction in QM, which now acts upon vacuum state. Then, in lagrangian of various quantum field theories, $\psi$ ...
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Dilation operator in CFT viewed as 'hamiltonian'?

From the commutation relations for the conformal Lie algebra, we may infer that the dilation operator plays the same role as the Hamiltonian in CFTs. The appropriate commutation relations are ...
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How to get the time derivative of an expectation value in quantum mechanics?

The textbook computes the time derivative of an expectation value as follows: $$\frac{d}{dt}\langle Q\rangle=\frac{d}{dt}\langle \Psi|\hat Q\Psi\rangle=\langle \frac{\partial\Psi}{\partial t}|\hat ...
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Angular momentum for 3D harmonic oscillator in two different bases

I know that the energy eigenstates of the 3D quantum harmonic oscillator can be characterized by three quantum numbers: $$ | n_1,n_2,n_3\rangle$$ or, if solved in the spherical coordinate system: ...
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About the definition of expectation value in quantum mechanics

In quantum mechanics, the expectation value of a observable $A$ is defined as $$\int\Psi^*\hat A\Psi$$ But in probability theory the expectation is a property of a random variable, with respect to a ...
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Are operators in quantum mechanics linear transformations?

Observables in quantum mechanics correspond to self-adjoint linear operators. If $\psi$ is an eigenvector of $\hat A$, then $\hat A\psi=\alpha\psi$ where $\alpha$ is the eigenvalue of $\psi$. ...
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The Eigenstate Existence Problem in Dirac's 'Principles of Quantum Mechanics'

In Chapter II of Dirac's Principles of Quantum Mechanics, Dirac explains that in general it is very difficult to know whether, for a given real linear operator, that any eigenvalues/eigenvectors exist ...
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What is the analogy of $|x\rangle$ in quantum field theory?

Let me start from path integral formulation in quantum mechanics and quantum field theory. In QM, we have $$ U(x_b,x_a;T) = \langle x_b | U(T) |x_a \rangle= \int \mathcal{D}q e^{iS} \tag{1} $$ ...
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What are the proper domains of the position and squared angular momentum operator?

I am looking at the position operator on a compact set $K \subset \mathbb{R}^n$ and the squared angular momentum operator (so essentially the Laplace-Beltrami operator where I just look at the angular ...
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Is the potential in Schrödinger equation an operator?

In the Schrödinger equation in the position representation $$ i\hbar\frac{\partial}{\partial t}\Psi(x,t) ~=~[\frac{-\hbar^2}{2m}\nabla^2+V(x,t)]\Psi(x,t), $$ is the potential $V(x,t)$ an operator ...
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Basis in quantum mechanics

My quantum mechanics textbook (Primer of Quantum Mechanics, by Marvin Chester) says that both the momentum space and the position space are basis spaces. It also says that the momentum space is ...
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Representation of operators in quantum mechanics

For which systems we represent the Hamiltonian as a differential operator and for which system we represent it by a matrix? Can the momentum be represented by a matrix operator?
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Position Representation in Quantum Mechanics

How does the 3d position operator look like in position representation? I know that in 1d the position operator $\hat{x}$ is just multiplication by $x$.
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Eigenstates of an observable

Can we use eigenstates of ANY observable as base of the Hilbert space? If we can, is this equal to the statement that those eigenstates are orthogonal to each other and normalizable?
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Why does the probability of obtaining a value of a measurement follow from Dirac's general assumption?

In Dirac's The Principle of Quantum Mechanics he makes the general assumption that "if the measurement of the observable $\xi$ for the system in the state corresponding to $|x\rangle$ is made a large ...
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Quantum Mechanics: Time dependence of an expectation value

In Griffiths's Introduction to Quantum Mechanics, he says that the time dependence of an expectation value is $$\frac{d\langle Q\rangle}{dt}=\frac{i}{\hbar}\langle [H,Q]\rangle+\langle \frac{\partial ...
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Operator Product Expansion in Massless 2D QED

In Peskin & Schroeder chapter 19 page 656, where the axial current anomaly of massless 2D QED is discussed, the authors go from: $$ ...
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Explanation of Dirac's proof of arbitrary ket being expressible with eigenkets of observable

In P.A.M. Dirac's The Principles of Quantum Mechanics, Chapter 10 (Observables), pp. 40, at the end of the chapter there is a proof that I don't understand at all. Here is a pdf link to the book ...
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Tricks at manipulating creation/annihilation operators

Manipulation of terms in algebras different from the standard one (e.g. boolean algebra) can be a bit unnatural but there are always shortcuts that can help you. I was wondering if there is a list ...
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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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How can I take the Wigner transform of an operator with an absolute value?

I want to be able to find the Wigner transforms of operators of the form $\Theta(\hat{O})$, where $\Theta$ is the Heaviside function and $\hat{O}$ in general depends on both $x$ and $p$. For the ...
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Role of physics in the zeta function $\zeta$ and the Riemann hypothesis

Hilbert and Polya suggested a physical way to verify the Riemann hypotesis about $\zeta(x)$. If the Riemann hypotesis is true, we can state all eigenvalues of physical problems are real. What is the ...
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Idempotent Operators

If $P$ is an idempotent operator, $P^2 = P$ and we have a vector $|\psi\rangle$, $P\neq1$, and the relation $$PL |\psi\rangle = L|\psi\rangle,$$ what conclusions can we draw about $L$, which is a ...
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Does Heisenberg equation of motion imply the Schrodinger equation for evolution operator?

Let us choose to postulate (e.g. considering the analogy of the Hamiltonian being a generator of time evolution in classical mechanics) $$ i\hbar \frac{d\hat{U}}{dt}=\hat{H}\hat{U}\tag{1} $$ where ...
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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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Deriving the expectation of $[\hat X,\hat H]$

For a free particle of mass $m$, with Hamiltonian $$\hat{H} = \frac {\hat{P}^2} {2m},$$ where $$\hat{P} = -i \hbar \frac{\partial} {\partial x}.$$ The commutative relation is given by $$[\hat{X}, ...
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What is the energy operator and from where do we get it?

I am trying to learn Quantum mechanics from MIT OCW Videos about quantum mechanics. I have reached the 5th lecture. Please help me in understanding this: In the middle (At 32:08), the professor wrote ...
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I am trying to calculate how $<r>$ in the hydrogen atom evolves with time

I am working on the Hydrogen atom and I was trying to calculate $\frac{d<r>}{dt}$ using $$\frac{d<r>}{dt} = \frac{i}{\hbar} <[\hat{H} , \hat{r}]>.$$ Here $r = \sqrt(x^2 + y^2 + z^2)$ ...
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Matrix elements of the operator $\hat{x} \hat{p}$ in position and momentum basis

I want to calculate the matrix elements of the operator $\hat{x} \hat{p}$ in momentum and position basis, that is the two quantities ($p,q$ - momenta, $x,y$ - positions): $$\langle p|\hat{x} ...
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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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Applications of the Spectral Theorem to Quantum Mechanics

I'm currently learning some basic functional analysis. Yesterday I arrived at the spectral theorem of self-adjoint operators. I've heard that this theorem has lots of applications in Quantum ...
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The notion of bounded states in quantum mechanics and their characterization with operators

Is there any case of potential $V$, such that the continuity of the operator $H=c\ \Delta+V$ is not spoiled? And I don't know any non-differnetial operator examples for continous spectra. I ...
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Bounded and Unbounded (Scattering) States in Quantum Mechanics

I understand that bounded states in quantum mechanics imply that the total energy of the state, $E$, is less than the potential $V_0$ at + or - spatial infinity. Similarly, the scattering state ...
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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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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 ...