Recent content by Thunder_Jet

Thanks for the detailed note. I did it but it turns out that the total integral vanish! What does it implies when the position representation is zero? I am expecting to get a Gaussian like solution. Or do you think I need to use Dirac delta function here instead of the exp(ipx/hbar) term?

Thanks for your suggestion. My problem now is on converting this momentum representation into its x representation. The probability density in x can be written as ∫<ψ(p)|x><x|ψ(p)> dx. Since I have here a complex conjugate of the Fourier transform term exp(ipx/hbar), those Fourier terms will...

Hi! I would like to ask everyone's opinion about this wavefunction in the momentum representation: ψ(p) = N[θ(-p)exp(ap/hbar) + θ(p)exp(-ap/hbar)], where N is a normalization constant, a > 0, and θ(p) is a function defined as θ(p) = 0 for p > 0 and also θ(p) = 0 for p < 0. I think the...

Hello! I am answering a problem which involves spins in the hamiltonian. The hamiltonian is given by H = B(a1Sz^(1) + a2Sz^(2)) + λS^(1)dotS^(2). The Sz^(1) and Sz^(2) refers to the Sz of the 1st and 2nd spins respectively. B is the magnetic field and the others are just constants. The...

Hmmm, sounds ok. Thank you for your suggestion. But I am really new to ladder operators, how would you use this translated a+ and a?

May I know how can I obtain the eigenvalues using the usual eigenvalue problem here? I am quite confused here now.

So that means just express the |n> kets as linear combinations of the ladder operators, and then use them as ψ in the formula <X> = <ψ|X|ψ>? But how would you deal with the infinite dimensionality? Will the answer be finite in that case? Thank you by the way for the idea!

Hi everyone! Given that a harmonic oscillator has eigenkstates |n> where n = 1,2,3,..., how can we calculate <X>, <P>, <X^2>, etc. Is there a need to define a wavefunction in the |n> basis? Thanks!

Ok, so a and a^+ are the annihilation and creation operators in the harmonic oscillator problem. I thought there are other operators. Thanks for your comment! Anyway, in this shifted harmonic oscillator case, do you expect that the solution for example, the eigenvalues are just shifted by a...

Hello again everyone! I would like to ask a question regarding this Hamiltonian that I encountered. The form is H = Aa^+a + B(a^+ + a). Then there is this question asking for the eigenvalues and ground state wavefunction in the coordinate basis. The only given conditions are, the commutator...

Thank you so much for the insights! All of your comments gave me an idea on how to attack the problem! Thanks once again!

Hi everyone! I would like to ask how would you verify if functions form an inner product space? For example, if one has functions of the form q(x)e^-(x^2/2) where q(x) is a polynomial of degree < N in x, on the interval -∞ < x < ∞. Also, how would you specify the dimension of the space, if it...

I see. So the problem is merely a Harmonic oscillator with shifted or rescaled x and p. Ok, I will try to arrange the eigenvalue equation to see the rescaling with respect to x and p. So the ground state eigenfunction and eigenvalue should be very similar to the original Harmonic oscillator...

A, I see. I think it's now a little bit manageable. It is actually similar to the differential equation for the Harmonic oscillator wherein the solutions are Hermite polynomials. Thank you very much for your time! Good night!

Yes, its possible that I just made some error, but the form is just the same. The problem now would be the differential equation that will arise from the Hamiltonian. Are there any special functions that will solve it or do I need to use methods such as power series, etc, or just introduce a...