Operator Definition and 1000 Threads

When I learned about operators, I learned <x> = ∫ Ψ* x Ψ dx, <p> = ∫ Ψ* (ħ/i ∂/∂x) Ψ dx. The book then told me the kinetic energy operator T = p2/2m = -ħ2/2m (∂2/∂x2) I am just think that why isn't it -ħ2/2m (∂/∂x)2 Put in other words, why isn't it the square of the derivative, but...

Hello, I'm trying to understand how to calculate de probability of finding a system in a specific eigenstate using the density operator. In the book of Balian, Haar, Gregg I've found a good definition of it being the expectation value of the projector Pr in the orientation of the eingenstate...

What mathematical operation does this expression represent" |i><i| I know the i's are unit vectors, and I know <i|i> is the dot product, but what operation is the ket-bra?

How do I know if some given state is and eigenstate of some given operator?

hi, initially I want to put into words that I looked up the link (http://physics.stackexchange.com/questions/86824/how-to-get-the-position-operator-in-the-momentum-representation-from-knowing-the), and I saw that $$\langle p|[\hat x,\hat p]|\psi \rangle = \langle p|\hat x\hat p|\psi \rangle -...

Hi For brevity one usually writes Fredholm integral equation of the 2nd kind $\psi(x)=f(x)+\int_{a}^{b} \,k(x,s)\psi(s) ds$ into the form $\psi=f+K \psi$ where $K$ is the operator kernel My question can one write an integro differential equation $\d{\psi(x)}{x}=f(x)+\int_{a}^{b}...

Hi. I'm confused about the action of the complex conjugate operator and time reversal operator on kets. I know K(a |α > ) = a* K | α > but what is the action of K on | α > where K is the complex conjugation operator ? What is the action of the time reversal operator Θ on a ket , ie. what is Θ...

How the authors came to the conclusion (eq. 4.25) that $$ U(t,t')=e^{iH_0(t-t_0)} e^{-iH(t-t')} e^{-iH_0(t'-t_0)} $$

Well i am noobie to quantum physics so i matbe totally incorrect so please bear with me. I had question how is position operator defined mathematically. I was reading the momentum position commutator from...

Homework Statement Hi, Just watching Susskind's quantum mechanics lecture notes, I have a couple of questions from his third lecture: Homework Equations [/B] 1) At 25:20 he says that ## <A|\hat{H}|A>=<A|\hat{H}|A>^*## [1] ##<=>## ##<B|\hat{H}|A>=<A|\hat{H}|B>^*=## [2] where ##A## and ##B##...

Homework Statement Eigenvalues of the Hamiltonian with corresponding energies are: Iv1>=(I1>+I2>+I3>)/31/2 E1=α + 2β Iv2>=(I1>-I3>) /21/2 E2=α-β Iv3>= (2I2> - I1> I3>)/61/2 E3=α-β Write the matrix of the Hamiltonian in the basis of...

hi, i want to ask you that how to perform an anti-difference operator on the product of two functions? i.e. D^{-1}(f_{n} g_{n})

1. Homework Statement prove the following statement: Hello, can someone help me prove this statement A is hermitian and {|Ψi>} is a full set of functions Homework Equations Σ<r|A|s> <s|B|c>[/B]The Attempt at a Solution Since the right term of the equation reminds of the standard deviation, I...

Homework Statement Vectors I1> and I2> create the orthonormal basis. Operator O is: O=a(I1><1I-I2><2I+iI1><2I-iI2><1I), where a is a real number. Find the matrix representation of the operator in the basis I1>,I2>. Find eigenvalues and eigenvectors of the operator. Check if the eigenvectors are...

https://wikimedia.org/api/rest_v1/media/math/render/svg/a7fd3adddbdfb95797d11ef6167ecda4efe3e0b9 https://en.wikipedia.org/wiki/Lorentz_force#Lorentz_force_in_terms_of_potentials How to write this formula in terms of sums and vector components? What is ##v\cdot\nabla## ? I think it is some...

In QFT, the commutation relation for the field operator \hat{\phi} and conjugate momentum is [\phi(x,t),\pi(y,t)] = i\delta(x-y) Maybe this is obvious, but what would the commutator of \phi or \pi and, say, e^{i k\cdot x} be?

The spin exchange operator would have the property $$\begin{align*}P\mid \chi_{\uparrow\downarrow} \rangle = \mid\chi_{\downarrow\uparrow} \rangle & &P\mid \chi_{\downarrow\uparrow} \rangle =\mid \chi_{\uparrow\downarrow} \rangle \end{align*}$$ This also implies ##P\mid \chi_{\text{sym.}}...

Hello :-) I have a small question for you :-) 1. Homework Statement The Operator e^{A} is definded bei the Taylor expanion e^{A} = \sum\nolimits_{n=0}^\infty \frac{A^n}{n!} . Prove that if |a \rangle is an eigenstate of A, that is if A|a\rangle = a|a\rangle, then |a\rangle is an...

http://<img src="https://latex.codecogs.com/gif.latex?L_{2}[0,1]->L_{2}[0,1]\int_{0}^{1}\left&space;|t-s&space;\right&space;|f(s)ds" title="L_{2}[0,1]->L_{2}[0,1]\int_{0}^{1}\left |t-s \right |f(s)ds" />[/PLAIN] I have many doubts on linear operator. How I can find a spectrum of a linear...

Hello, When we normalize the free particle by putting it in a box with periodic boundary conditions, we avoid the "pathological" nature of the momentum representation that take place in the normal problem of a particle in a box with the usual boundary conditions of Ψ=0 at the two borders. Thus...

Hi. I have come across the following equation < x | p | ψ > = - iħ d/dx < x | ψ > where p is the momentum operator. I'm confused as to how p which acts to the right on | ψ >can then be taken outside the inner product so it now also acts on the bra < x | ? Thanks

I know the positive field operator E+ is actually an annihilation operator a while the negative field E- is a creation operator a+. I also learned that the absorption process can be represented as E-E+, which should be the number of photons n accroding to the principle of ladder operator. Also...

I can derive $$a(k) = \int d^3 x e^{ik_{\mu} x^{\mu}} (\omega_{\vec{k}} \psi + i \pi)$$ for a free real scalar Klein-Gordon field in three ways mathematically: the usual Fourier transform way in Peskin/Srednicki, an awesome direct a = ½(2a) = ... way (exercise!), and as a by-product of a clever...

While studying Ehrenfest's theorem I came across this formula for time-derivatives of expectation values. What I can't understand is why is position/momentum operator time-independent? What does it mean to be a time-dependent operator? Since position/momentum of a particle may change...

Homework Statement This isn't really a problem. I am just re-reading some section "Classical Mechanics" by John Taylor. I think this belongs in the math section, since my question is mainly about the del operator. There is just one fragment of one sentence that I want to make sure I am...

I don't understand what the eigenvalue of a translation operator means physically. The eigenvalues of other operators like momentum and hamiltonian give us the physically measurable values I suppose. Then what exactly do we obtain by the translation eigenvalues? I am new to the field of quantum...

How can I show (without using truth table) that p → q is equivalent to F ↓ ((F ↓ p) ↓ q) where F is constant "false" and p and q are propositions? Is it possible to have a similar kind of expression with T (true) instead of F? Thanks in advance!

Homework Statement [/B] For a general operator ## \hat{O}##, let ##\hat{O}_{mn}(t)## be defined as: $$ \hat{O_{mn}}(t) = \int u^{*}_{m}(x,t) \hat{O} u_{n}(x,t) $$ and $$ \hat{O_{mn}} = \int u^{*}_{m}(x) \hat{O} u_{n}(x) $$ ##u_{m}## and ##u_{n}## are energy eigenstates with corresponding...

In Carroll, the author states: \nabla^{\mu}R_{\rho\mu}=\frac{1}{2} \nabla_{\rho}R and he says "notice that, unlike the partial derivative, it makes sense to raise an index on the covariant derivative, due to metric compatibility." I'm not seeing this very clearly :s What's the reasoning...

With an example, can someone explain me what the following code would do? In fact I am not certain I understand how the |= operator is used. (I have checked for operator overloading within the classes but there is none, so it supposedly does the default operation) objectA = ClassA(arguments)...

Hello, Could someone provide me with a good proof or explain me here how we can derive Slater determinant for N fermions by starting with the vacuum state and the creation operators with anticommutation equations. I see that the idea of both these structures is similar but I cannot work it out...

Hi, I stumbled upon thinking that "Is curl operator a linear operator" ? I was reading EM Theory and studied that the electromagnetic field satisfies the curl relations of E and B. But if the operator was not linear then how can a non linear operator give rise to a linear solution. Thus it...

Homework Statement I am watching a course on Relativistic Quantum Mechanics to freshen up, and I have found to have some issues regarding simple operator algebra. This particular issue on the Pauli Equation (generalization of the Schrodinger equation that includes spin corrections) in an...

Hi, quick question with A being the lowering operator and A† the raising operator for a QHO (A A† - 1 + 1/2) ħω [Aψ] = A (A† A - 1 + 1/2) ħω ψ By taking out a factor of A. Why has the ordering of A A† swapped around? I would have thought taking out a factor of A would leave it as A (A† - 1 +...

How can I prove that the eigenvalues of the operator ## i\gamma^\mu \partial_\mu ## are non-negative? I've tried using the ansatz ## \psi=u(p) e^{ip_\nu x^\nu} ## but it didn't help. I've also tried playing with the equation using the properties of gamma matrices but that doesn't seem to lead...

Hey there! 1. Homework Statement I've been given the operators a=\sqrt\frac{mw}{2\hbar}x+i\frac{p}{\sqrt{2m\hbar w}} and a^\dagger=\sqrt\frac{mw}{2\hbar}x-i\frac{p}{\sqrt{2m\hbar w}} without the constants and definition of the momentum operator: a=x+\partial_x and a^\dagger=x-\partial_x with...

Hello, i am kind of confused about something. What is the complex conjugate of the momentum operator? I don't mean the Hermitian adjoint, because i know that the Hermitian adjoint of the momentum operator is the momentum operator. Thanks!

Homework Statement A gaussian wave packet is given by the formula: Ψ(x)=(1/(π1/4d1/2))eikx-(x2/2d2) Calculate the expectation value in this quantum state of the momentum squared. Homework Equations <p2>=-ħ∫Ψ*(X) (d2Ψ(x)/dx2) dx ∫e(-x2/d2) dx= d√π ∫xe(-x2/d2) dx =0 ∫x2e(-x2/d2) dx = (d3√π)/2...

The problem is actually of an introductory leven in Quantum Mechanics. I am doing a course on atomic and molecular physics and they wanted us to practice again some of the basics. I want to know where I went conceptually wrong because my answer doesn't give a total probability of one, which of...

This question was inspired by 3c) on https://people.phys.ethz.ch/~muellrom/qm1_2012/Solutions4.pdf Given the operator \hat{B} = \left(\matrix{b&0&0\\0&0&-ib\\0&ib&0}\right) I find correctly that the eigenvalues are \lambda = b, \pm b. To find the eigenvectors for b, I do the following...

Hi. I have come across the following step in a derivation of the harmonic oscillator groundstate wavefunction using ladder operators ∫ <x | p | p><p | o > dp = ∫ p<x | p><p | o > dp = -iħ d/dx ∫ <x | p><p | 0>dp I am confused about how the -iħ d/dx arises. I thought the p produced when the p...

Hello everyone, I have a question. I'm not sure if it is trivial. Does anyone have ideas of finding a matrix ##A\in M_n(\mathbb{C})##, where ##A## is normal but not self-adjoint, that is, ##A^*A=AA^*## but ##A\neq A^*?##

Hi everyone, Weinberg uses spatial translation invariance to derive the momentum operator. But the way he does it puzzles me. Here is an excerpt of the book. Equation 3.5.1 is the definition of the unitary operator ##U(x)## for translation invariance: $$U^{-1}(x)XU(x) = X+x,$$ with...

I saw a link on MIT open courseware proving grad operator yields perpendicular vector to contour, but I can't make sense of how dg/dt=0. Can someone explain to me please...

Homework Statement Prove the following relations (for ##\zeta:=r e^{i\theta}##): \begin{align} D(\alpha)^\dagger a D(\alpha)&=a+\alpha\\ S(\zeta)^\dagger a S(\zeta)&= a \cosh r- a^ \dagger e^{i\theta} \sinh r \end{align} Homework Equations ##|\alpha\rangle## is the coherent state. ##a## and...

The formula T = -(ħ/2m)∇2 implies that T is proportional to the second spatial derivative of a wavefunction. What is the origin of this dependence? In classical mechanics, T = p2/2m. Is it also the case in classical mechanics that p2/2m is proportional to a second spatial derivative? I...

Hi all! I'm having problems understanding the operator algebra. Particularly in this case: Suppose I have this projection ## \langle \Phi_{1} | \hat{A} | \Phi_{2} \rangle ## where the ##\phi's ## have an orthonormal countable basis. If I do a state expansion on both sides then I suppose I'd get...

hi, am major new on quantum mechanics. please help me understand. is the real wave function Ψ2Px= [Ψ2p+1 +Ψ2p-1]1/2 an eigen function of L2 or Lz? if so, how is it? and if so kindly explain the values of l and m thanks

In the Griffiths textbook for Quantum Mechanics, It just gives the ladder operator to be L±≡Lx±iLy With reference to it being similar to QHO ladder operator. The book shows how that ladder operator is obtained, but it doesn't show how angular momentum operator is derived. Ive searched the...

In the Hamilonian for an H2+, the kinetic energy of the electron (KE of nucleus ignored due to born-oppenheimer approximation) has a negative sign in front of it. I understand the signs for the potential energy operators but not for the KE apart from the strictly mathematical point of view. Can...