Operator Definition and 1000 Threads

Note: physics conventions, \theta is measured from z-axis We have a vector operator \vec{L} = -i \vec{r} \times \vec{\nabla} = -i\left(\hat{\phi} \frac{\partial}{\partial \theta} - \hat{\theta} \frac{1}{\sin\theta} \frac{\partial}{\partial \phi} \right) And apparently \vec{L}\cdot\vec{L}=...

Homework Statement Hi, it's me again. I'm new to vector calculus so this might sound like a stupid question, but in relation to a specific problem, I was wondering when we could move the del operator under the integration sign - in relation to a specific problem, which is: A(r) = integral...

Prove that if operator on a hilbert space $T$ commutes with an operator $S$ and $T$ is invertible, then $T^{-1}$ commutes with $S$. $T^{-1}S$=$T^{-1}T^{-1}TS$=$T^{-1}T^{-1}ST$

Let $T_{n}$ be a sequence of invertible bounded linear operators with limit $T$ Prove that $(T_{n})^{-1}$ tends to $T^{-1}$

Homework Statement This is a university annual exam question: Show that for a potential V (-r)=-V (r) the wave function is either even or odd parity. Homework Equations The Attempt at a Solution We can determine whether a wavefunctions' parity is time independent based on if the...

Homework Statement A particle is in a 1D harmonic oscillator potential. Under what conditions will the expectation value of an operator Q (no explicit time dependence) depend on time if (i) the particle is initially in a momentum eigenstate? (ii) the particle is initially in an energy...

Hi guys, this might be a stupid question but if I wanted a general expression for the time evolution of the angular momentum operator is it just the same as Hamiltonian? i.e ih ∂/∂t ψ = L2 ψ Solving this partial differential gives the time evolution of the angular momentum operator...

Typically in mathematics time derivative is linear in the sense that constants are pulled out the operator which then operates on a time dependent function. But in quantum mechanics we say linear to mean that the operator passes over the coefficients of the kets (which themselves might be time...

What are the "matrix elements" of the angular momentum operator? Hello, I just recently learned about angular momentum operator. So far, I liked expressing my operators in this way: http://upload.wikimedia.org/math/8/2/6/826d794e3ca9681934aea7588961cafe.png I like it this way because it...

I am feeling a little stupid tonight... So let me build the problem... For a single particle operator O, we have in the basis |i> we have that: O= \sum_{ij} o_{ij} |i><j| with o_{ij}=<i|O|j> Then for N particles we have that: T=\sum_{a}O_{a}= \sum_{ij} o_{ij} \sum_{a} |i>_{a}<j|_{a} with...

Homework Statement \hat T = \frac{{\hat L_z^2}}{{2I}} = - \frac{{{\hbar ^2}}}{{2I}}\frac{{{\partial ^2}}}{{\partial {\varphi ^2}}} Homework Equations Find eigenfunctions and eigenvalues of this operatorThe Attempt at a Solution It leads to the differential eqn - \frac{{{\hbar...

Homework Statement Im having a hard time figuring out how in quantum mechanics things such as momentum can be expressed as an operator. I know the simple algebra to get the relation. Starting with the 1D solution to wave equation\Psi=e^{i\omega x} then differentiating that with respect to x...

Homework Statement I'm running through practice papers for my 3rd year physics exam on atomic and nuclear physics: This is the operator we found in the previous part of the question L = -i*(hbar)*d/dθ Next, we need to find the eigenvalues and normalised wavefunctions of L The...

If g(a) \neq 0 and both f and g are continuous at a, then we know the quotient function f/g is continuous at a. Now, suppose we have a linear operator A(t) on a Hilbert space such that the function \phi(t) = \| A(t) \|, \phi: \mathbb R \to [0,\infty), is continuous at a. Do we then know that...

Homework Statement I have problem where I need to commutate my hamiltonian H with a fermionic anihillation operator. Had H been written in terms of fermionic operators I would know how to do this, but the problem is that it describes phonon oscillations, i.e. is written in terms of bosonic...

There is something I do not understand. One way to define the current density operator is through the particle density operator Ï(r). From the fundamental interpretation of the wavefunction we have: Ï(r)= lψ(r)l2 But my book takes this a step further by rewriting the equality above...

I am studying how to express Greens functions in imaginary time formalism. I have however big problems understanding the attached derivation. In equation 10.15 and 10.16 in...

Hi everyone. I am studying 'identical particles' in quantum mechanics, and I have a problem with the properties of the Symmetrizer (S) and Antisymmetrizer (A) operators. S and A are hermitian operators. Therefore, for what I know, their set of eigenkets must constitute a basis of the space...

Is it possible to take momentum operator as dr/dt (r is position operator)? If not, why?

I am learning quantum mechics. The hypothesis is: In the quantum mechanics, all operators representing observables are Hermitian, and their eigen functions constitute complete systems. For a system in a state described by wave function ψ(x,t), a measurement of observable F is certain to...

Suppose I have this operator: ##D^2+2D+1##. Is the ##1## there, when applied to a function, considered as identity operator? Say: ##f(x)=x##. Applying the operator results in: ##D^2(x)+2D(x)+(x)## or ##D^2(x)+2D(x)+1##? If ##1## here is considered as an identity operator then the...

Hello, I'm solving the previous exams and I have a problem with an exercise: Homework Statement q(x) a real function defined in [0,1] and continuous L a sturm Liouville operator : Lf(x)=f''(x)+q(x)*f(x) f ∈ C²([0,1]) with f(0)=0 and f'(1)=0. Is L a symetric operator relative to the...

I have started coming across square roots (H+kI)^{\frac 12} of slight modifications of Schrodinger operators H on L^2(\mathbb R^d); that is, operators that look like this: H = -\Delta + V(x), where \Delta is the d-dimensional Laplacian and V corresponds to multiplication by some function. But...

Attached is a derivation of the equation of motion for the fundamental fermionic anihillation opeator but I am having a bit of trouble with the notation. Does the notation Vv2-v and the other V_ simply mean that all terms in the sum of q have canceled except for when q=v2-v or v-v1? And second...

If I haven't understood this tricky stuff very badly when the Hamiltonian is time independent, then Schrödinger’s equation implies that the time evolution of the quantum system is unitary, but for the time-dependent Hamiltonian one must add some mathematically "put by hand" assumptions (although...

Hi folks, originally I read Peskin & Schroeder but then I realized it was too concise for me. So I switched to Srednicki and am reading up to Chapter 5. (referring to the textbook online edition on Srednicki's website) Two questions: 1. In the free real scalar field theory, the creation...

Homework Statement Let V be a finite-dimensional vector space over F, and let T : V -> V be a linear operator. Prove that T is indecomposable if and only if there is a unique maximal T-invariant proper subspace of V. Homework Equations The Attempt at a Solution I tried using the...

Hello, Suppose P is a projection operator. How can I show that I+P is inertible and find (I+P)^-1? And is there a phisical meaning to a projection operator? (Please be patient I have just started with QM). Thanks. Y.

I really asked this question in another thread but it seems the original respondent gave up explaining me. My question is about the rewriting of the integrals from first to second line on the attached picture. The θ denotes the heaviside step function such that: θ(t1-t2) = {1 t1>t2 , 0 t1<t2}...

My tutor showed me something today, and I still can't completely wrap my head around why it makes sense. Consider the following integral equation: ##\int f(x) = f(x) - 1## Then: ##\int (f(x)) - f(x) = -1 \implies f(x) \left( \int (\text{Id}) - 1\right) = -1## so we get the geometric series...

I'm having trouble seeing how this equality is possible which is seen in proofs of properties of normal operators. ||Tv||^2 = <T*Tv, v> = <TT*v, v> = ||T*v||^2 As far as I can get is ||Tv||^2 = <Tv, Tv>

Homework Statement Show there are no eigenvectors of a^{\dagger} assuming the ground state |0> is the lowest energy state of the system. Homework Equations Coherent states of the SHO satisfy: a|z> = z|z> The Attempt at a Solution Based on the hint that was given (assume there...

For the time evolution operator: exp(-iHt) How do I take the derivative of an operator like this keeping the order correct? I mean I of course know how to differentiate an exponential function, but this is the exponential of an operator.

Homework Statement . Let ##X=\{f \in C[0,1]: f(1)=0\}## with the ##\|x\|_{\infty}## norm. Let ##\phi \in X## and let ##T_{\phi}:X \to X## given by ##T_{\phi}f(x)=f(x)\phi(x)##. Prove that ##T## is a linear continuous operator and calculate its norm. The attempt at a...

Given that D²f(x) = g(x), one form that eliminate the second derivate is integrating the equation: ∫∫D²f(x)dx² = ∫∫g(x)dx². But, and if I try so: \\ \sqrt{D^2f(x)}=\sqrt{g(x)} \\ D\sqrt{f(x)}=\sqrt{g(x)} \\ PD\sqrt{f(x)}=P\sqrt{g(x)} \\ \sqrt{f(x)}=P\sqrt{g(x)} \\ f(x)=[P\sqrt{g(x)}]^2 \\...

I have an operator which is written in k space as something like: H = Ʃkckak where a_k and c_k are operators. So it is a sum of operators of different k but there are no crossterms as you can see. Being no crossterms does this mean that the operator is diagonalized in the language of linear...

A group ##G## is said to act on a set ##X## when there is a map ##\phi:G×X \rightarrow X## such that the following conditions hold for any element ##x \in X##. 1. ##\phi(e,x)=x## where ##e## is the identity element of ##G##. 2. ##\phi(g,\phi(h,x))=\phi(gh,x) \ \ \forall g,h \in G##. My...

Hi guys, I need help on interpreting this solution. Let me have two wave functions: \phi_1 = N_1(r) (x+iy) \phi_2 = N_2(r) (x-iy) If the angular momentum acts on both of them, the result will be: L_z \phi_1 = \hbar \phi_1 L_z \phi_2 = -\hbar \phi_2 My concern is, \phi_1 and \phi_2...

Particle Number Operator (Hermitian??) Hey guys, I'm studying the quantic harmonic oscillator and I'm using "Cohen-Tannoudji Quantum Mechanics Volume 1". At some point he introduced the particle number operator, N, such that: N=a+.a , where a+ is the conjugate operator of a. The...

Hi everyone, :) Here's a problem that I want to confirm my answer. Note that for the second part of the question it states, "prove that \(T\) is bonded by the above claim". I used a different method and couldn't find a method that relates the first part to prove the second. Problem: Suppose...

My friends and I have been struggling with the following problem, and don't understand how to do it. We have gotten several different answers, but none of them make sense. Can you help us? **Problem statement:** Let $V$ be the vector space of real-coefficient polynomials of degree at most $3$...

Homework Statement Given the Hamiltonian H(t) = \frac{P^2}{2m} + \frac{1}{2}mw^2X^2 + b(XP+PX) from some b>0. Find an annihilation operator a_b s.t. [a_b,a_b^{\dagger}]=1 and H = \hbar k (a_b^{\dagger}a_b+\frac{1}{2}) for some constant k. Hint: [P + aX,X]=[P,X], \forall a.Homework Equations...

Hello everyone I've got these things buzzing in my head and not exactly knowing how to solve them. Homework Statement Operator Ahat = (d/dx + x) and Bhat = (d/dx - x) a. Chat = AhatAha b. Chat = AhatBhat What do the position and momentum operator Xhat = x and Phat = -i*hbar*d/dx, give when...

1. What are the possible eigenvalues of the spin operator \vec{S} for a spin 1/2 particle? Homework Equations I think these are correct: \vec{S} = \frac{\hbar}{2} ( \sigma_x + \sigma_y + \sigma_z ) \sigma_x = \left(\begin{array}{cc}0 & 1\\1 & 0\end{array}\right),\quad...

Greetings, I must be missing something obvious but how is Tr{} defined exactly in case of contunuous spectrum operators? Everywhere I look I see it defined as a sum of [possibly infinite sequence of] eigenvalues. Is the following correct: Given Q = \int f(q) \left| q\right\rangle...

Hey guys, So this question is sort of a fundamental one but I'm a bit confused for some reason. Basically, say I have a Hermitian operator \hat{A}. If I have a system that is prepared in an eigenstate of \hat{A}, that basically means that \hat{A}\psi = \lambda \psi, where \lambda is real...

Hi, Could anyone tell if there exists an identity a_k^\dagger = a_{-k} because intuitively there should be no difference between creating a particle with momentum k and destroying a particle with momentum -k. If true is it possible to show that from the definition a_k = \frac{1}{√V}∫e^{ikx}...

Homework Statement Hey guys, I'll type this thing up in Word. http://imageshack.com/a/img716/8219/wycz.jpg

I have been trying to convert the Del operator from Cartesian to Cylindrical coords since like 5 days. but still i can't see why my way doesn't work. It worked for the 3D heat equation and 3D wave equation but for vector quantities no :( ... This is the way i followed \nabla P =...

Homework Statement Consider a particle whose wave function is: \Psi(x)=\left\{\begin{array}{ccc} 2\alpha^{3/2}xe^{-\alpha x} & \text{if} & x> 0\\ 0 & \text{if} & x\leq 0 \end{array}\right. Calculate <p> using the \hat{p} operator on probability density in x space. Homework...