Recent content by WisheDeom

Homework Statement Prove that ##\lim_{x \rightarrow a} f(x) = \lim_{h \rightarrow 0} f(a+h)##. Homework Equations By definition, if ##\lim_{x \rightarrow a} f(x) = l## then for every ##\epsilon > 0## there exists some ##\delta_1## such that for all x, if ##0<|x-a|<\delta_1## then...

I know there are a number of tests for series of numbers, but I'm not sure how to translate this to operators. My class didn't do any rigorous operator calculus; we sort of played it by ear. But in this case I'm not even sure how to start really. If I assume it does converge, the sum should be...

Homework Statement I must prove that the set of coherent states \left\{ \left| \lambda \right\rangle \right\} of the quantum simple harmonic oscillator (SHO) is a complete set, i.e. it forms a basis for the Hilbert space of the SHO. Homework Equations The coherent states are defined as...

I think you meant [U^n,x], and if so, I got it! Thanks a lot.

Ah! Thank you, that was straightforward.

Homework Statement Let \left|x\right\rangle and \left|p\right\rangle denote position and momentum eigenstates, respectively. Show that U^n\left|x\right\rangle is an eigenstate for x and compute the eigenvalue, for U = e^{ip}. Show that V^n\left|p\right\rangle is an eigenstate for p and...

Quantum Theory: Commutators of Functions of Observables Homework Statement First is a question from Sakurai Modern Quantum Mechanics, 2nd ed., 1.29a. Show that [x_i,G(\mathbf{p})] = i\hbar\frac{\partial G}{\partial p_i} and [p_i,F(\mathbf{x})] = - i\hbar\frac{\partial...

If a differential equation allows for multiple solutions, the most general solution may contain an arbitrary term. This is always the case for linear equations, for example. If, in addition, boundary conditions are given, then the arbitrary constant is set by those conditions. As a basic...

I don't have much to say on the content, but some friendly advice: Try breaking your OP into paragraphs. I bet more people will respond if you make it a little more readable. :smile:

Ah! I understand now. So by definition A \cap B = \left\{x:x \in A \wedge x \in B } and C \subseteq A \leftrightarrow \forall x (x \in C \rightarrow x \in A). Since the intersection of A and B contains elements from A but no elements not in A, (A \cap B) \subseteq A. Is this valid? Does...

Thank you for the responses. Yes, of course. This is what I was trying to do in the second part, but see below. Would you be able to go into a little more detail? I know this is very basic, but I am just starting. :smile: Yeah, that's what I meant. I guess my approach in learning...

Hello, I am teaching myself Set Theory, and in doing some exercises I came across the problem: Given sets A and B, prove that A \subseteq B if and only if A \cap B = A. My proof, in natural language, is in two parts: 1) Prove that if A \subseteq B, A \cap B = A. By the definition...