Operators Definition and 1000 Threads

Suppose we have a Hamiltonian containing a term of the form where ∂=d/dr and A(r) is a real function. I would like to study this with harmonic oscillator ladder operators. The naïve approach is to use where I have set ħ=1 so that This term is Hermitian because r and p both are.*...

Ref. 'Core Principles of Special and General Relativity' by Luscombe. Apologies in advance for the super-long question, but it's necessary to show my thought process. Let ##\gamma:I\to M## be a smooth curve from an open interval ##I\subset\mathbb{R}## to a manifold ##M##, and let...

Good morning! I have a problem in understanding the steps from vectors to operators. Imagine you are given a vectorial observable. In classical mechanics, after rotating the system it transform with a rotation matrix R. If we go to quantum mechanics, this observable becomes an operator that is...

I just realized quantum operators X and P aren't actually just generalizations of matrices in infinite dimensions that you can naively play with as if they're usual matrices. Then I learned that the space of quantum states is not actually a Hilbert space but a "rigged" Hilbert space. It all...

Consider the field creation operator ψ†(x) = ∫d3p ap†exp(-ip.x) My understanding is that this operator does not add particles from a particular momentum state. Rather it coherently (in-phase) adds a particle created from |0> expanded as a superposition of momentum eigenstates states...

The magnitude of the momentum ##p## satisfies ##p^2 = p_x^2 + p_y^2 + p_z^2## and this implies the operator equation ##\hat{p}^2 = \hat{p}_x^2 + \hat{p}_y^2 + \hat{p}_z^2##, so we can say that ##\hat{p}^2 = -\hbar^2 (\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} +...

"Given a 3D Harmonic Oscillator under the effects of a field W, determine the matrix for W in the base given by the first excited level" So first of all we have to arrange W in terms of the creation and annihilation operator. So far so good, with the result: W = 2az2 - ax2 - ay2 + 2az+ 2 -ax+...

IfA=cd, where c and d are annihilation operators of two different types of Fermions, then {A,A°}is? A.1+n1+n2 B.1-n1+n2 C.1-n2+n1 D.1-n1-n2 Where,n1 and n2 are corresponding number operator, A° means A dagger or creation operator,as the particles are fermions they will obey anti-commutation I think

In theory, does an algebraic expression exist for the ground state of the Klein Gordon equation with \phi^4 interactions in the same way an algebraic expression exists for the simple harmonic oscillator ground state wavefunction in Q.M.? Is it just that it hasn't been found yet or is it...

I've recently stumbled upon something that looked kind of silly, but I still find myself a bit confused by it. Namely in quantum field theory, when we quantize a scalar field, we impose commutation relations on creation and annihilation operators that correspond to momenta in their mode...

Let's go step by step a) We know that the harmonic oscillator operators are $$a^{\dagger} = \frac{1}{\sqrt{2 \hbar m \omega}} ( -ip + m \omega q)$$ $$a= \frac{1}{\sqrt{2 \hbar m \omega}} (ip + m \omega q)$$ But these do not depend on ##L##, so I guess these are not the expressions we want...

I found a theorem that states that if A and B are 2 endomorphism that satisfies $$[A,[A,B]]=[B,[A,B]]=0$$ then $$[A,F(B)]=[A,B]F'(B)=[A,B]\frac{\partial F(B)}{\partial B}$$. Now I'm trying to apply this result using the creation and annihilation fermionics operators $$B=C_k^+$$ and $$A=C_k$$...

I am getting started in applying the quantization of the harmonic oscillator to the free scalar field. After studying section 2.2. of Tong Lecture notes (I attach the PDF, which comes from 2.Canonical quantization here https://www.damtp.cam.ac.uk/user/tong/qft.html), I went through my notes...

I know that ahat_+ = 1/sqrt((2*m*h_bar*w)) * (mw(xhat)+i(phat)) and ahat_- = 1/sqrt((2*m*h_bar*w)) * (mw(xhat)-i(phat)). But I'm not sure what (ahat_+)^+ could be.

Hello, I am struggling with what each piece of these equations are. I generally know the two rules that need to hold for an operator to be linear, but I am struggling with what each piece of each equation is/means. Lets look at one of the three operators in question. A(f(x))=(∂f/∂x)+3f(x) I...

Given a wave function \Psi which is an eigenstate of a Hermitian operator \hat{Q}, we can measure a definite value of the observable corresponding to \hat{Q}, and the value of this observable is the eigenvalue Q of the eigenstate $$ \hat{Q}\Psi = Q\Psi $$ My question is whether it's a postulate...

Li=1/2*∈ijkJjk, Ki=J0i,where J satisfy the Lorentz commutation relation. [Li,Lj]=i/4*∈iab∈jcd(gbcJad-gacJbd-gbdJac+gadJbc) How can I obtain [Li,Lj]=i∈ijkLk from it?

Why is it that the raising and lowering operators in a spin 1/2 system have a factor of $\hbar ?$ From Sakurai: $$S_+ \equiv \hbar | + \rangle \langle - |, S_- \equiv \hbar | - \rangle \langle + |$$ "So the physical interpretation of $S_+$ is that it raises the component by one unit of $\hbar...

Dear PF, I hope I've formulated my question understandable enough. Thank you for your time, Garli

Does the time ordering operator ##\mathcal{T}## commute with the normal ordering operator ##\hat{N}##? i.e. is $$[ \mathcal{T},\hat{N}] =0$$ correct?

While deriving continuity equation in Fluid mechanics, our professor switched the order of taking total time derivative and then applying delta operator to the function without stating any condition to do so(Of course I know it is Physics which alows you to do so) . So,I began to think...

Hello everyone, I have noticed a striking similarity between expressions for creation/annihilation operators in terms position and momentum operators and trigonometric expressions in terms of exponentials. In the treatment by T. Lancaster and S. Blundell, "Quantum Field Theory for the Gifted...

I have always learned that a Hermitian operator in non-relativistic QM can be treated as an "experimental apparatus" ie unitary transformation, measurement, etc. However this makes less sense to me in QFT. A second-quantised EM field for instance, has field operators associated with each...

I'm unclear on what exactly an annihilation or creation operator looks like in QFT. In QM these operators for the simple harmonic oscillator had an explicit form in terms of $$ \hat{a}^\dagger = \frac{1}{\sqrt{2}}\left(- \frac{\mathrm{d}}{\mathrm{d}q} + q \right),\;\;\;\hat{a} =...

Can somebody provide an explanation why the dynamical variables/observables are represented in QM as linear operators with the measured values being eigenvalues of these operators? For energy this is probably trivially and directly follows from the stationary Shrodinger equation which solutions...

I'm trying to understand some notes that I have been given on Matrix Mechanics, specifically how the matrix element comes about and builds a matrix which when used applies the effect of an operator on a wavefunction. But I'm having some difficulties following what's being done in the notes with...

I am reading Zettili’s “Quantum Mechanics: Concepts and Applications” and I am in the section on functions of operators. It starts with how ##F(\hat A)## can be Taylor expanded and gives the particular and familiar example: $$e^{a \hat A} = \sum_{n=0}^\infty \frac{a^n}{n!} \hat A^n...

Hello! In several of the derivations I read so far in my QFT books (M. Schawarz, Peskin and Schroeder) they use the fact that "we can safely assume that the fields die off at ##x=\pm \infty##" in order to drop boundary terms. I am not sure I understand this statement in terms of QFT. A field in...

Assume ##P_1## and ##P_2## are two projection operators. I want to show that if their commutator ##[P_1,P_2]=0##, then their product ##P_1P_2## is also a projection operator. My first idea was: $$P_1=|u_1\rangle\langle u_1|, P_2=|u_2\rangle\langle u_2|$$ $$P_1P_2= |u_1\rangle\langle...

I started and successfully showed that the expectation of X_1 and X_2 are zero. However the expectation value of X1^2 and X2^2 which I am getting is <X1^2> = 0.25 + \alpha^2 and <X2^2> = 0.25. How do I derive the given equations?

a. ##L_{+}^{\dagger}=(L_x+iL_y)^{\dagger}=L_x-iL_y=L_{-}## b.##[L_{+},L_{-}]=[L_x+iL_y,L_x-iL_y]=(L_x+iL_y)(L_x-iL_y)-(L_x-iL_y)(L_x+iL_y)=## ##=L_x^2-iL_xL_y+iL_yL_x+L_y^2-(L_x^2+iL_xL_y-iL_yL_x-L_y^2)## ##=L_x^2-iL_xL_y+iL_yL_x+L_y^2-L_x^2-iL_xL_y+iL_yL_x+L_y^2##...

Let's consider Bohm's paradox (explaining as follows). A zero spin particle converts into two half-spin particles which move in the opposite directions. The parent particle had no angular momentum, so total spin of two particles is 0 implying they are in the singlet state. Suppose we measured Sz...

In Principles of Quantum mechanics by shankar it is written that Pi is a projection operator and Pi=|i> <i|. Then PiPj= |i> <i|j> <j|= (δij)Pj. I don't understand how we got from the second result toh the third one mathematically.I know that the inner product of i and j can be written as δijbut...

I'm studying how derivatives and partial derivatives transform under a Galilean transformation. On this page: http://www.physics.princeton.edu/~mcdonald/examples/wave_velocity.pdf Equation (16) relies on ##\frac{\partial t'}{\partial x}=0## but ##\frac{\partial x'}{\partial t}=-v## But this...

http://vergil.chemistry.gatech.edu/notes/quantrev/node20.html "Postulate 2. To every observable in classical mechanics there corresponds a linear, Hermitian operator in quantum mechanics. " "Postulate 6. The total wavefunction must be antisymmetric with respect to the interchange of all...

That's my attempting: first I've wrote ##e## in terms of the power series, but then I don't how to get further than this $$ \sum_{n=0}^\infty (-1)^n \frac {Â^n} {n!} \hat B \sum_{n=0}^\infty \frac {Â^n} {n!} = \sum_{n=0}^\infty (-1)^n \frac {Â^2n} {\left( n! \right) ^2} $$. I've alread tried to...

Are there new hermitian operators in quantum gravity? Background: In many worlds interpretation (MWI). We have the preferred basis problem and the basis are for example position, momentum, spin. Each of those bases come from a hermitian operator: they are the eigenbasis of the (for example)...

Homework Statement From Griffiths GM 3rd p.266 Consider a free particle of mass ##m##. Show that the position and momentum operators in the Heisenberg picture are given by$$ {\hat x}_H \left( t \right) ={\hat x}_H \left( 0 \right) + \frac { {\hat p}_H \left( 0 \right) t} m $$ $$ {\hat p}_H...

Dear all, I've been reading and got confused of the concept below have two questions question 1) For <ψ|HA|ψ> = <Hψ|A|ψ>, why does the Hamiltonian operator acting on the bra state and <ψ|AH|ψ> in this configuration it will act on the ket state? question 2) what does it mean for H|ψ> = |Hψ>...

Hi, I'm starting to study how to use Hubbard operators and I cannot understand one property: Consider the hopping terms for a lattice Hamiltonian with bosons: $$\sum_{i,j\neq i} t_{i,j} b^\dagger_i b_j$$ when writing this term in the basis of Hubbard operators $$X^{a,b}_i =| a,i \rangle \langle...

When we make a symmetrie transformation in a quantum system, the state ##|\psi \rangle## change to ## |\psi' \rangle = U|\psi \rangle##, where ##U## is a unitary or antiunitary operator, and the operator ##A## change to ##A'##. If we require that the expections values of operators don't change...

If A and B are Hermitian operators is (i A + B ) a Hermitian operator? (Hint: use the definition of hermiticity used in the vector space where the elements are quadratic integrable functions) I know an operator is Hermitian if: - the eigenvalues are real - the eigenfunction is orthonormal -...

Homework Statement Given the expression s_{\pm}|s,m> = \hbar \sqrt{s(s+1)-m(m\pm 1)}|s,m \pm 1> obtain the matrix representations of s+/- for spin 1/2 in the usual basis of eigenstates of sz Homework Equations s_{\pm}|s,m> = \hbar \sqrt{s(s+1)-m(m\pm 1)}|s,m \pm 1> S_{+} = \hbar...

Cohererent states are defined as eigenstates of the annihilation operator. Never the creation operator is referred to. Is this just a convention or is more behind? What is the essential difference between eigenstates of the annihilation- versus the creation operator? Thank you very much in...

Hi! When calculating ##(\hat{a} \hat{a}^{\dagger})^2## i get ##\hat{a} \hat{a} \hat{a}^{\dagger} \hat{a}^{\dagger}## which is perfectly fine. But how do I end up with the ultimate simplified expression $$\hat{ a}^{\dagger} \hat{a} \hat{a}^{\dagger} \hat{a} + \hat{a}^{\dagger} \hat_{a} +...

One way to get the gradient of polar coordinates is to start from the Cartesian form: ##\nabla = \hat x \frac{\partial}{\partial x} + \hat y \frac{\partial}{\partial y}## And then to use the following four identies: ##\hat x = \hat r\cos\theta - \hat{\theta}\sin\theta## ##\hat y = \hat...

Hi! I am working on homework and came across this problem: <n|X5|n> I know X = ((ħ/(2mω))1/2 (a + a+)) And if I raise X to the 5th, its becomes X5 = ((ħ/(2mω))5/2 (a + a+)5) What I'm wondering is, is there anyway to be able to solve this without going through all of the iterations the...

I started watching the video lecture series here: https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/video-lectures/part-1/ I notice that they use the term "operator" without first explaining it. Operators are also not explained (in fact they are not even mentioned) in my...

Hi, It appears that the definition of a quantum field creation operator is given by $$\Psi^{\dagger}(\mathbf r) = \sum\limits_{\mathbf k} e^{-i\mathbf k\cdot \mathbf r} a^{\dagger}_{\mathbf k}.$$ But then if we examine how this operator acts on the vacuum state, we get $$\Psi^{\dagger}(\mathbf...

Homework Statement Consider two operators A and B, such that [A,[A, B]] = 0 and [B,[A, B]] = 0 . Show that Exp(A+B) = Exp(A)Exp(B)Exp(-1/2 [A,B]) Hint: define Exp(As)Exp(Bs) as T(s), where s is a real parameter, differentiate T(s) with respect to s, and express the result in terms of T(s)...