Recent content by Calu

Homework Statement Let the commutator [A,B] = cI, I the identity matrix and c some arbitrary constant. Show [A,Bn] = cnBn-1 Homework Equations [A,B] = AB - BA The Attempt at a Solution So I have started off like this: [A,Bn] = ABn - BnA = cI I'm not sure where to go from here.

Homework Statement Set |Ψ> = (1/√ 2) | ↑, z> + (eiθ / √ 2) | ↓, z>. Find the probability of measuring the spin component of sz to be up the z-axis. Find the probability of measuring the spin component of sx to be up the x axis. Homework Equations I'm not sure. The Attempt at a Solution I...

Thanks very much for your help.

I think this is where I'm confused: In the first part there is |L - f(z*)| < ε, whereas you've written that is has to be the case that |f(z*) - L| < ε. Are the two equivalent? I see now how |f(z*) - l'| < ε and |f(z*) - l'| < ε must be true as they come from the definition of the limit. However...

Sorry, I thought I'd deleted part of the quote. This is the part I'm confused about: I realize that they must be smaller than ε, I just don't see how they are.

That is the part I'm confused on. I realize that they must be smaller than ε, I just don't see how they are.

Homework Statement I'm struggling with the proof that the limit of a complex function is unique. I'm struggling to see how |L-f(z*)| + |f(z*) - l'| < ε + ε is obtained. Homework Equations 0 < |z-z0| < δ implies |f(z) - L| < ε, where L is the limit of f(z) as z→z0 .The Attempt at a Solution...

Homework Statement We have n ≥ 2, n not prime, n ∈ ℤ. Take the smallest such n. n is not prime and as such n is not irreducible and can be written as n = n1.n2; n1, n2 not units. We may take n1, n2 ≥ 2. However we have n > n1, n > n2 so n1, n2 have prime factors. I'm not sure how n > n1, n >...

I see now, thanks for your help.

Right I see, this would give me what I have in the second sum, obviously with m substituted for n. So can this change of variable be used as I'm summing over twice as many integers?

Homework Statement I have that, for n ∈ ℕ, (-6/π) ∑n even (cos(nx))/(n2-9) is equivalent to (-6/π) ∑∞n=1 (cos(2nx))/(4n2-9). I don't understand how the two sums are equivalent to each other. Homework Equations I honestly have no idea what may be relevant, other than what is above. The...

I know that the magnitude of ##\sqrt{n+1}## is larger than that of ##\sqrt{n}##. Therefore I would assume that the opposite would be true for the magnitude of their reciprocals which would make an≥an+1 as required, however I'm not sure how to write this in a more coherent way.

Homework Statement I was wondering how I would go about showing that (an) is monotone decreasing given that an = 1/√n. I believe I have to show an ≥ an+1, but I'm not sure how to go about doing that.

I see, thank you very much.

Homework Statement I have been asked to prove the convergence or otherwise of ∑∞n=1 n/(3n + n2). In the example solution, with the aim to prove divergence by comparison with the Harmonic Series, the lecturer has stated that n/(3n + n2) ≥ n/(4n2) = 1/4n and which diverges to +∞. I was...