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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Conjugate of an operator applied to a function

In section 6.3.1 of the following MIT Open Course Ware (PDF) (22-02, Introduction to Applied Nuclear Engineering), the author, Prof Paola Cappellaro, derives Heisenberg's equation using the definition ...
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90 views

Unitary transformation between complete + orthonormal bases

Suppose the complete orthonormal bases $\{|\psi_n\rangle\}$ and $\{|\psi{'}_n\rangle\}$ are related by the transformation matrix $U$: $$ |\psi{'}_n\rangle = U|\psi_n\rangle \\ \langle\psi{'}_n| = ...
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176 views

Expectation Values and Derivation of Heisenberg Equation?

Consider a system of particles with wave function $\psi$(x) (x can be understood to stand for all degrees of freedom of the system; so, if we have a system of two particles then x should represent ...
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112 views

Apply the Heisenberg Equation to the Hamiltonian [closed]

$\frac{d}{dt}$$\hat{H}$ = $\frac{i}{\hbar}$$[\hat{H},\hat{H}]$ +$\frac{\partial{\hat{H}}}{\partial{t}}$ That's as far as I've got. I do not know much about the Heisenberg equation or even what it ...
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35 views

Uncertainty Definition QM

On my introductory course in Quantum Mechanics, the uncertainty of an operator $A$ in the state $\psi$ is defined by $$(\Delta A)^2_{\psi}=\langle(A-\langle A \rangle_{\psi})^2\rangle _{\psi}$$ I'm ...
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Quantum Expectation Values

I'm having trouble understanding the motivation for the definition of the expectation of a self adjoint operator $A$: $$\langle A \rangle _\psi=\int_{\mathbb{R}}\psi^*A\hspace{0.2cm} \psi ...
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88 views

Commutator summation notation

I have the relation $ e^L M e^{-L}=\sum_{n=0}^\infty \frac 1{n!} [L,M]_{(n)}$ where $L$ and $M$ are operators. What does the subscript $n$ after the commutator bracket denote?
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349 views

Non-Hermitian operator with real eigenvalues?

So we know that in Quantum Mechanics we require the operators to be Hermitian, so that their eigenvalues are real ($\in \mathbb{R}$) because they correspond to observables. What about a non-Hermitian ...
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Question about equation 2.27 from Pachos's Introduction to topological quantum computing

http://quince.leeds.ac.uk/~phyjkp/Files/IntroTQC.pdf above is the PDF that is hosted on his website. The equation is on page 22 (pg 30 in the pdf). In chapter 2. It is the second equation of the ...
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Showing that the maximum possible uncertainty for any observable is half the difference between its maximum and minimum eigenvalues

Show that the maximum possible uncertainty for any observable is $\frac{1}{2}|x_2 - x_1|$ where $x_1$ and $x_2$ are the extreme eigenvalues of X (Maximize $\Sigma_i p_ix_i^2 - (\Sigma_i p_ix_i)^2$) ...
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86 views

Momentum and position operators in Schrödinger representation

I was going through some intro notes on path integral (for QFT), and am stuck with this equation for position and momentum in Schrödinger (position) representation, $$ \hat{1} =\int ...
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145 views

Self-adjointness

I know I have posted this question before some time ago. But no one could help so I decided to put my problem in another background. The Schrödinger equation of a free scalar field is given by ...
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41 views

Renormalizing with external momenta set to zero

I've often seen in textbooks that authors renormalize diagrams by setting external momentum to zero. Under what conditions is this justified? An example of this is done in Manohar and Wise's book on ...
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55 views

Commutators of differential and field operators

In $p+ip$ superconductivity, a term in the Hamiltonian (in polar coordinates) is $$ H=\int \mathrm{d}^2\vec{r} \left[ \left( \partial_r -\frac{i}{r} \partial_{\theta} \right) \psi^\dagger(\vec{r}) ...
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211 views

Expectation Value of a Dynamical Variable

In quantum mechanics, we generally take about "expectation values of dynamical variables". However, by the postulates of quantum mechanics, every dynamical variable in quantum theory is represented by ...
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Help understanding proof in simultaneous diagonalization

The proof is from Principles of Quantum Mechanics by Shankar. The theorem is: If $\Omega$ and $\Lambda$ are two commuting Hermitian operators, there exists (at least) a basis of common eigenvectors ...
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342 views

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 the expectation value of the square of momentum be negative?

I've been solving a problem in quantum mechanics, and I was deriving the standard deviation of $P$, knowing that $\langle P\rangle=0$. Because $\Delta P=\sqrt{\langle P^2 \rangle - \langle P \rangle ...
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92 views

Momentum Operator in Quantum Mechanics

1) What is the difference between these two momentum operators: $\frac{\hbar}{i}\frac{\partial}{\partial x}$ and $-i\hbar\frac{\partial}{\partial x}$? How are these two operators the same? My ...
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119 views

Comparing two infinite sets

All the linearly independent eigenfunctions of the parity operator $\mathcal{P}$ form an infinite set and all the linearly independent eigenfunctions of the unit operator $\bf 1$ also form an ...
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203 views

Matrix elements of linear operators - orthonormal basis required?

In an early linear algebra class of mine, I learnt that a linear map $\mathcal{A}$ acting on a vector space could be represented by a matrix $A_{ij}$ according to the rule: $$\mathcal{A}({e_j}) = ...
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142 views

Precise meaning of composition of ket and bra, e.g. $|\psi\rangle\langle\psi|$

I'm currently studying density matrices, and have been frequently coming across the construction $$|\psi\rangle\langle\psi| \,.$$ What is the formal meaning of this composition? I understand ...
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Why don't we use Hamilton-Jacobi method in QM?

In classical mechanics, we usually try to find a set of coordinates by Hamilton-Jacobi method to transform the Hamiltonian to zero such that the coordinates are conservations. However, we never try ...
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160 views

Difference between expectation value and probability amplitude?

I was given a wave equation. I know that probability amplitude is the eigenvalue of an observable operating in a state. $$H| \psi\rangle = h| \psi\rangle$$ where $h$ is the probability amplitude of ...
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How does $\bar{r}\times(\bar{\nabla}\times) - \bar{\nabla}\times(\bar{r}\times)$ relate to the orbital angular momentum operator?

When I attempted to calculate the following by hand $$\bar{r}\times(\bar{\nabla}\times\bar{F}) - \bar{\nabla}\times(\bar{r}\times\bar{F}),$$ I noticed some of the terms I extracted looked similar to ...
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The formal solution of the Schrodinger equation

Let's have Schrodinger equation (or some equation in Schrodinger form) $$ \tag 1 i \partial_{0} \Psi ~=~ \hat{H} \Psi . $$ One likes to write that it has formal solution $$ \tag 2 \Psi (t) ~=~ ...
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What is the commutator of an operator and its derivative?

Is it possible to calculate in a general way the commutator of an operator $O$ which depends on some variable $x$ and the derivative of this $O$ with respect to $x$? $${O}={O}(x)\\ ...
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75 views

Chirality and helicity

I have massless Dirac equation and chirality and helicity operators which are given as $$ \hat {P}_{ch}\Psi = \gamma_{5}\Psi, \quad \hat {P}_{h}\Psi = \frac{(\hat {\mathbf S} \cdot \mathbf ...
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232 views

Writing an arbitrary operator in bra-ket notation

An annoying fact about my physics textbook (Griffiths' Introduction to Quantum Mechanics) is that it introduces bra-ket notation without telling us how to use it. So I have a two-part question for SE: ...
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1answer
84 views

How to find different operator representations in QM?

I read that any observable operator may be represented as: $$\Omega = \sum_n \omega _n | \omega _n \rangle \langle \omega_n |$$ Where the little omegas are the eigenvectors/eigenvalues of the ...
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461 views

Self-adjoint and unbounded operators in QM

An operator $A$ is said to be self-adjoint if $(\chi,A\psi)=(A\chi,\psi)$ for $\psi, \chi \in D_A$ and $D_A=D_{A^\dagger}$. But for the free particle momentum operator $\hat{p}$ these inner products ...
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How is the set of displacement operators best called?

Displacement operators $\hat D(x,p), \ \ x,p\in\mathbb{R},$ follow a composition rule $$D(x,p) D(x',p') = \exp\frac{i(px'-xp')}2 D(x+x',p+p').$$ Because of the extraneous phase factor, the set of all ...
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Position and momentum bases in quantum mechanics

I have seen the following two descriptions of the position basis: $$\tag{1}| x\rangle=\delta(x-x_0)$$ and also $$\tag{2}\langle x_0| x\rangle=\delta(x-x_0),$$ which (if either) of these is ...
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How can (in Dirac's terminology) the product of two “real” linear operators be “not real”?

I'm puzzled about a statement from Dirac's book, The principles of quantum mechanics, (§8, p.28): As a simple examples of this result, it should be noted that, if $\xi$ and $\eta$ are real, in ...
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292 views

The Holstein-Primakoff Representation (approximation)

I have a question regarding the Holstein-Primakoff representation. In the HP-representation we define the spin operators in terms of bosonic creation and annihilation operators. $$ S_j^+ = \sqrt{2S ...
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1answer
96 views

Normal ordering of the identity operator

I'm puzzled about what should be the normal ordering of the identity operator (or any proportional operator): looking at it from the "Fock space operators POV",the prescription is to move all the ...
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1answer
58 views

Electric field in a cylinder

We have electric charge density $\rho(r) = kr$ in a cylinder of infinite height and radius $a$. I'm asked to find the electric field. I'm doing it using two methods and I don't undesrtand why then ...
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Virasoro Operators commutation relations

For the commutation relation in quantising the bosonic string $\left[L_n,L_{m}\right]=(n-m)L_{n+m}+\frac{D}{12}n(n^2-1)\delta_{n+m,0}$ we can then calculate this for $m=-n$ in between the vacuum ...
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174 views

Intrinsic parity of particle and antiparticle with spin zero

I need to prove that the intrinsic parities of a particle and antiparticle with spin zero are the same. Can I prove that by an argument that operator of $P$-inversion commutes with charge conjugation ...
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71 views

About Divergence in polar coordinates

I've got a conductor in a cylinder shape who is rotating with angular velocity $\omega$ around its axis, that correspond to the $z$ azis I want to calculate the electric field and the density of ...
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1answer
82 views

Eigenfunction associated with the $\hat{x}$ operator

Consider the following operator $\hat{x}=i\hbar \frac{\partial}{\partial p}$. I am trying to show that the eigenfunctions of $\hat{x}$ are not square-normalizable. I am interested in doing so since ...
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166 views

Shortcut to find $\hat{p}^2$ expectation value

I have been going through several calculations where I am asked to calculate $\langle p^2 \rangle$ and the task is proving to be pretty tedious. Does anyone know of a shortcut for this? Such as with ...
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1answer
58 views

Any other bound state problems using $a_+$ and $a_-$?

Why is it that creation and annihilation operators ($a_+$ and $a_-$) can only be defined for the problem of quantum harmonic oscillator and nothing else? Can any other bound state problem be solves ...
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77 views

Translation Operator for Position on Momentum

Consider the translation operator $$\hat T=\exp[-ic\hat p/\hbar], $$ which acts on the position operator in the following way: $$\hat T^\dagger \hat q\hat T = \hat q+c.$$ If I take $\hat T ^\dagger ...
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Bounded and Unbounded Operator

Can someone explain with a concrete example of how can I can check whether a quantum mechanical operator is bounded or unbounded? EDIT: For example., I would like to check whether $\hat ...
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78 views

Self-adjoint and nonpositive differential operators

I recently tumbled over a statement in a geophysics paper (PDF here). They have a wave equation which they formulate as $$ \frac{1}{v_0}\frac{\partial^2}{\partial t^2} \begin{pmatrix}p \\ ...
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64 views

Translation Operator on two operators

On my last HW set, we were asked to show that the operator $$\hat T = \exp(-ic\hat p /\hbar)$$ acts as a translation operator ($\hat T^\dagger q\hat T=q+c)$. This was simple to show using commutators ...
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293 views

Self-adjoint differential operators

I'm having a hard time understanding the deal with self-adjoint differential opertors used to solve a set of two coupled 2nd order PDEs. The thing is, that the solution of the PDEs becomes ...
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170 views

Chronological and normal ordering

I've realized I'm little bit confused when I want to treat elements like this $$\left<\phi_0|T\{a_p(t)a_p^+(t')V(t_1)V(t_2)\}|\phi_0\right>$$ with $$V(t)=\dfrac12 \dfrac{1}{(2\pi ...
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1answer
135 views

Wick's theorem for calculating OPE

I am trying to understand a calculation using Wick's theorem. Let $T(z)$ be the analytic part of a stress-energy tensor, and $\phi(z)$ a free boson field. Now, ...