Show that the minimum energy of a simple harmonic oscillator is fw/2 if ArAp = h/2, where (Ap)² = ((p - (p))?). %3D %3D
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- 1) Consider a trial wavefunction $(r) = N e-r for the estimation of the ground state energy of the hydrogen atom. (a) Calculate the variational energy W[ø] using the trial wavefunction 6(r). (b) To obtain the best result (that is, the one that is closest to the true ground state energy) minimize your result with respect to the parameter b. (c) How does your result in (b) compare with E1, the ground state of the hydrogen atom. Explain.explain the Fermi Dirac distributionif ; ψ=(2a/π)^1/4*exp(-ax^2) what are the <x^2>=? ,<p^2>=?
- The product of the two provided equations (with Z = 1) is the ground state wave function for hydrogen. Find an expression for the radial probability density and show that it is maximized at r = a0.We are going to use Heisenberg's uncertainty principle to estimate the ground- state energy of hydrogen. In our model, the electron is confined in a one- dimensional well with a length about the size of hydrogen, so that Ax = 0.0529 nm. Estimate Ap, and then assume that the ground-state energy is roughly Ap2/2me. (Give your answer in Joules or electron-volts.)The product of the two provided equations (with Z = 1) is the ground state wave function for hydrogen. Find an expression for the radial probability density and show that the expection value for r (for the ground state) is <r> = 3a0/2.
- An electron is in a 3p state in the hydrogen atom, given that the expectation value is 12.5a_0 What is the probability of finding the electron within +/- a_0 of your expectation value. (That is, in the range (r − a_0) < r < (r+a_0) where r is the expectation value from above. The answer should be 0.1991.Show that the uncertainty principle can be expressed in the form ∆L ∆θ ≥ h/2, where θ is the angle and L the angular momentum. For what uncertainty in L will the angular position of a particle be completely undetermined?Show that the following function Y(0,9)= sin 0 cos e eiº is the solution of Schrödinger 1 1 equation: sin 0 21 sin 0 00 Y(0,0)= EY (0,9) and find the sin 0 dp? energy, E.
- Show that the radial wave function R21 for n = 2 and ℓ = 1 is normalized.da=do= 4) For the following d hydrogenic wavefunctions, find the magnitude and the z component of the orbital angular momentum. You can give your results in terms of h. اور اہل = 1 √2 16t i√2 i√2 Final (d₁2+ d_2) = (165) R₁2(r) (x²- y²)/r² 151/2 1/2 R₁2(r) (3cos²0-1)=(16) R2(1)(32²-7²)/² 1/2 (d+2-d-2)= (15) Rm2(7)xy/² =(d. 1+d-1)= (d_1-d-1)=- 1/2 1/2 (15) R₁2 (r)yz/r² 10 June 2011 1/2 (45) * Ru2(1) 2x/r²An H2 molecule can be approximated by a simple harmonic oscillator with a force constant k = 1.1 x 103 N/m. Find (a) the energy levels and (b) the possible wavelengths of photons emitted when the H2 molecule decays from the third excited state eventually to the ground state.