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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Proof that a Hermitian Matrix is not defective?

I am taking an introductory course into Quantum Mechanics. To me to seems pretty simple to prove most properties of Hermitian operators. However, I am stuck at an edge case, proving that if an ...
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63 views

QFT: Ground State Momentum - Normalisation of States

In my notes I have, $$ \left\langle \mathbf{p} \left| \mathbf{q} \right.\right\rangle = \left\langle 0 \left| {a(\mathbf{p})}\ {a(\mathbf{q})}^{\dagger} \right| 0 \right\rangle $$ I am not sure how ...
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100 views

Conservation of momentum in infinite square well

This is inspired by Griffiths QM section 2.2, on the infinite square well, which is about how far I've gotten (so, sorry if this is addressed later in the book). For any given starting wavefunction, ...
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72 views

Commutation of two vector operator [closed]

Consider vectors $\overrightarrow { A } $ and $\overrightarrow { B } $ as operators or vector of operators. If this commutation holds$$[\overrightarrow { A },\overrightarrow { B }]=0$$ Then, is that ...
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1answer
101 views

Galilean relativity in QM

Intro I've been trying to show that the generator of boosts can be written in operator form as can be seen here, as: $$ B = \sum_i m_i x_i(t) - t \sum_i p_i $$ As a reminder the transformation ...
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Why do $\hat{X}$ and $\hat{P}$ have to correspond to position and momentum?

As far as I understand, in QM we treat observables as operators, and the eigenvalues of these operators are the possible values we can measure of the observables. It is usually simpler to work in the ...
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96 views

Manipulation of operators in quantum mechanics

I'm reading some notes on quantum mechanics that state the following. $$\langle x\rvert \left( \hat{x} + \frac{i\hat{p}}{m\omega}\right) \lvert E \rangle = 0 \Rightarrow \left( x+ \frac{\hbar}{m\...
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1answer
72 views

Why aren't $\hat{x}$ and $\hat{p}$ considered functions of time in the expectation value?

In Griffiths Intro to QM (2nd edition), he gives the equation $$ \frac{d}{dt} \langle Q \rangle = \frac{i}{\hbar}\langle [\hat{H},\hat{Q}] \rangle + \left\langle \frac{\partial{\hat{Q}}}{\partial{t}}...
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23 views

Why should an generator acts on an operator with the Lie bracket?

When we deal with ordinary symmetries which form a Lie group, we have an corresponding Lie algebra with a structure of Lie bracket $[,]$. A infinitesimal transformation can act on a state or an ...
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1answer
134 views

Kronecker sum or direct sum?

When we write $$H=\sum_k H_k$$ in condensed matter physics, are we using Kronecker sum or direct sum? I think this is direct sum. However, Wikipedia says it is Kronecker sum. Can anyone give some ...
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84 views

Vector operators in quantum mechanics

Vector operators $\vec{V}$ in quantum mechanics are usually defined as those that commute in a particular way with the spatial Angular Momentum $\vec{L}$: $[L_i,V_j]=i\hbar\varepsilon_{ijk}V_k$. I am ...
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1answer
80 views

Proof that trace is independent of representation [closed]

$$\begin{align} \sum_{a'} \langle a'|X|a'\rangle &=\sum_{a',b',b''} \langle a'|b'\rangle \langle b'|X|b''\rangle\langle b''|a'\rangle \\ &=\sum_{b',b''} \langle b''|b'\rangle \langle b'|X|b''\...
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1answer
54 views

Lorentz covariant completeness relation

Let be $$ P^\mu |p> = p^\mu |p> $$ i.e. $|p>$ is the eigen-vector of the 4-momentum operator. Where does the following Lorentz-covariant completeness relation come from? $$ \int d^4p \...
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38 views

Determine adjoint operator [closed]

I need to determine the adjoint operator to $a_i$: $a_i |n_0, n_1, ... \rangle_S = \sqrt{n_i} |n_0, n_1, ..., n_{i-1}, n_i - 1, n_{i+1}, .. \rangle_S$ where the $S$ should denote the symmetrizer. I ...
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35 views

Matrix representation of a fermionic creation and annihilation operator in graphene nanoribbons?

From the other question Matrix representation for fermionic annihilation operator, what if we have to find the matrix representation for the operators $a_{\sigma}^{\dagger}(k,n)$ and $b_{\sigma}(k,n)$ ...
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1answer
45 views

Heisenberg picture transition amplitudes

I want to calculate the transition amplitude for a particle to start at position $q_1$ at time $t_1$ to position $q_2$ at time $t_2$ in the Heisenberg picture. As we are in the Heisenberg picture, ...
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Baryonic operators in ${\cal N}=1$ $U(N)$ SQCD in four dimensions

Seiberg's duality is usually considered as a duality for $SU(N_c)$ theories with $N_f$ flavors. In his case, the vacuum for $N_f \geq N_c$ is parameterized by mesons $M$ and baryons ${\bar B}$ and $B$....
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1answer
46 views

Why is the angular momentum added for two independent electron system? (no problem)

There is no problem now. But somebody may be confused by the same analysis when studying QM or Group theory. (actually my motivation for asking this question comes from the SU(5) Grand Unification ...
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2answers
90 views

Negative powers of operators

This may sound like a strange question, but just to be sure: Suppose I have a general Hermitian operator in Hilbert space whose action on an eigenvector is given by $R|r\rangle = r|r\rangle$. Then, I ...
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135 views

Radial quantum number for infinite circular well

For completeness, I will sketch the solution of a particle in an infinite circular well first and then get to my question. I apologize in advance since the introduction is standard undergraduate ...
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1answer
61 views

Limit of the position and momentum commutator [closed]

The commutator of position and momentum operator, $\hat{p}$ and $\hat{x}$, respectively is derived as $[\hat{x},\hat{p}]=i\hbar$. Let $\lim_{x\rightarrow x_{o}} [\hat{x},\hat{p}]=\lim_{x\rightarrow x_{...
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50 views

Spin-1/2 rotation operator: rotation about an angle of $\pi$

The spin-1/2 rotation operator: $$ R_{n}(\alpha) = \begin{pmatrix} cos(\frac{\alpha}{2})-in_{z}sin(\frac{\alpha}{2}) & (-in_{x}-n_{y})sin(\frac{\alpha}{2}) \\ (-in_{x}+n_{y})sin(\frac{\...
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1answer
156 views

Is the energy always discrete?

In the von Neumann axioms for quantum mechanics, the first postulate states that a quantum state is a vector in a separable Hilbert space. It means it is assumed the Hilbert space has a basis with at ...
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1answer
60 views

Spectrum of Laplacian on one hemisphere

as is well-known, the spectrum of the Laplace operator on $S^2$, computed via $-\Delta f=\lambda f$, is positive and discrete. What happens to the spectrum if we just take one hemisphere into account?...
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37 views

Lorentz invariance of matrix element of Heisenberg operator

The following text is taken by Weinberg book of QFT Volume 1, pg.437 Let's consider $O_l(x)$ an Heinsenberg-picture operator with the Lorentz transformation properties of some sort of free field $\...
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32 views

Can we create ladder operators for any potential? [duplicate]

I understand the mechanism for the harmonic oscillator, so can we generalize the ladder operator in order to work for any 1-D symmetric potential such as Gaussian potential well?
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2answers
143 views

Are the eigenstates of an operator time independent?

In the Schrodinger picture, are the eigenstates of an operator time independent? Is it their expectation values that evolve in time rather than the actual eigenstates? For example, say I have an ...
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3answers
148 views

Prove time-dependent hamiltonian is hermitian from unitarity of time-evolution operator

When we solve the Schrodinger equation for the time-evolution operator: \begin{equation} i\hbar\frac{\partial}{\partial t}U(t,t_{0})=HU(t,t_{0}), \end{equation} We have three cases to be treated ...
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0answers
50 views

The commutator between observable and unit radius vector

As I encounter the commutator relating to unit radius vector, I am quite confused. I have just started the learning of quantum mechanics and all I know about the commutator is based on two identities: ...
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54 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 $\langle x | p\rangle$...
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124 views

Why Use the Non-Relativistic Momentum Operator in Relativistic Quantum Mechanics?

In deriving the Klein Gordon equation one starts out with the relativistic energy relation E^2 = p^2 + m^2 and substitutes the quantum momentum operator that corresponds to non-relativistic QM, p = -i ...
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1answer
94 views

Confusion with time ordering

I am thinking about Proof of correlation function formula in quantum field theory and have realized there is a deeper confusion underpinning that. Consider: $$T\{U_I(T, t_2)\Phi_I(x_1)\}$$ where $...
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1answer
87 views

Clarification in deriving the radial momentum operator $p_r$

In deriving an expression for $p_r$, a particle's radial momentum, I am unsure what is happening at a certain step. The derivation given in The Physics of Quantum Mechanics by Binney and Skinner is as ...
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1answer
177 views

Recovering QM from QFT

Reading through David Tong lecture notes on QFT. On pages 43-44, he recovers QM from QFT. See below link: QFT notes by Tong First the momentum and position operators are defined in terms of "...
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1answer
134 views

Time-ordered product vs path integral

Suppose we have the Green function $$ G(k) \equiv \tag 1\int d^4x e^{ikx}\langle 0| T\left(\partial^{x}_{\mu}A^{\mu}(x)B(0)\right)|0\rangle , $$ which in path integral approach is equal to $$ \tag 2 ...
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228 views

Derivation of canonical position-momentum commutator relation

We know that the position-momentum commutator is fundamental in quantum mechanics, but would it be possible to derive it starting from a different set of first principles, more specifically starting (...
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96 views

Fermion Lagrangian with linear momentum versus quadratic momentum

$$ L = \bar{\psi} (\gamma^\mu (p_\mu -A_\mu)- m)\psi \tag{1} $$ $$ L = \bar{\psi} ((\gamma^\mu( p_\mu-A_\mu))^2 - m^2)\psi \tag{2} $$ Is there a difference between the two Lagragians in equations 1 ...
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1answer
67 views

Definition of leading log terms in one loop corrections for gravity

One loop corrections for gravity usually includes non-local terms in the action such as $R\log(\frac{-\Box}{\mu^2})R$, where $\Box=g^{\mu\nu}\nabla_\mu\nabla_\nu$ is the D'Alembert operator, $R$ is ...
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132 views

Interpretation of $\langle \phi | A | \psi \rangle$ [duplicate]

If the current state of some quantum system is $| \psi \rangle$, what is the physical interpretation of $$ \langle \phi | A | \psi \rangle $$ where $|\phi\rangle$ is some other -maybe the same- ...
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1answer
50 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 ...
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1answer
97 views

Operator algebra for momentum and magnetic vector potential

Let $\vec{A}$ be the magnetic vector potential and $\vec{p}$ be momentum. $$ \vec{p} \cdot \vec{A} \psi = (\vec{p} \cdot \vec{A}) \psi + \vec{A} \cdot (\vec{p} \psi) $$ $$ \vec{A} \cdot \vec{p} \psi =...
2
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1answer
123 views

In the algebraic formulation of Quantum Mechanics, how do probability amplitudes naturally arise?

In the algebraic formulation of quantum mechanics, consider $\mathcal{B}(\mathcal{H})$ as the set of all bounded operators on $\mathcal{H}$ (with involution, norm, etc.), which form a C*-algebra $C$. ...
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1answer
46 views

Expectation to uncertainty

We know that in the case of $O$ being an operator, $\langle O^2\rangle-\langle O\rangle^2$ equals to uncertainty as long as $\langle\rangle$ means the mean value (expectation value). if we have $A$ ...
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988 views

Can momentum have a complex expectation value?

I'm making examples of wave functions to incorporate in a QM exam. I came up with the following wave function, which gives me some troubles: $$\psi(x,0) = \begin{cases} A(a-x), & -a \leq x \leq a\...
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566 views

Mutual or same set of eigenfunctions if two operators commute

If two operators commute, do they have "a mutual set of eigenfunctions", or "the same set of eigenfunctions"? My quantum chemistry book uses these as if they are interchangeable, but they do not seem ...
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33 views

Spin operators in quantum mechanics

I'm reading for my exam on Monday. In my notes I have written that the teacher has told us that: $S_1 \cdot S_2 = [(S_1 + S_2)^2 - S_1^2 - S_2^2]/2 = S_{tot}^2 - \frac{3 \hbar^2}{4}$ Where $S_i = (...
1
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1answer
135 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 ...
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2answers
297 views

Is there a time operator in quantum mechanics?

The question in the title has been asked many times on this site before, of course. Here's what I found: Time as a Hermitian operator in QM? in 2011. Answer states time is a parameter. Is there an ...
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1answer
104 views

Inserting the resolution of identity correctly

In a text on path integrals, I find the following: \begin{equation} \langle q_{j+1}|e^{-i(\hat{p}/2m)\delta t}|q_j\rangle = \int\frac{dp}{2\pi}\langle q_{j+1}|e^{-i(\hat{p}/2m)\delta t}|p\rangle\...
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163 views

Perturbation Theory Question

How can you work out the average perturbation, from a normal hamiltonian, of all states that rely on the quantum numbers of s = __ and l = __, with the perturbation being proportional to the product ...