17. The Hamiltonian for the H-atom is shown below, where Kh...

17. The Hamiltonian for the H-atom is shown below, where $K_h$ is a constant. \hat{H} = \frac{-K_h}{2} \left[ \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2}{\partial \phi^2} \right] - \frac{K_h}{r} a) Use this Hamiltonian to calculate the energy of the 1s orbital: $\Psi_{1s} = Ne^{-r}$ b) Now calculate the energy of the 2p$_z$ orbital: $\Psi_{2pz} = Nr \cos \theta \cdot e^{-r/2}$ c) And for the brave, calculate the energy of the 2p$_x$ orbital: $\Psi_{2px} = Nr \sin \theta \cdot e^{-r/2} \cdot e^{i\phi}$

Transcript

00:01 In this problem, we are given a quantum mechanical system and indeed the radial part of the hamiltonium for the hydrogen atom as follows.

00:14 H equal to minus 1 over 2 r squared, d over d r, inside the derivative r squared, d over d r, minus 1 over r r.

00:28 And this hamiltonian is expressed in atomic units.

00:32 Now we are going to determine or compute the energy of the ground state using a trial function where we are going to employ the variational method.

00:47 So we have this trial function given by e to the power minus alpha r and here alpha is a variation parameter.

00:59 And with the trial function, the ground state energy can be estimated by, let's call this e the ground state energy.

01:12 The expectation value of the hamiltonian in this state divide by the normalization of this state.

01:22 Since most probably our trial wayfunction is not normalized, so we should always divide our expectations values by this normalization factor over there.

01:38 Now that's said, this is the formalism we are going to use.

01:42 So let's start calculating these objects, both the normalization and the expectation value of the hamiltonian.

01:53 I will start normalization because that's easier.

01:57 We have the entire 3d volume integral of si squared, i mean the absolute square of this sci function, the trial way function.

02:17 So the volume element in three dimensions in spherical coordinates is given by, and i'm considering spherical coordinates because our trial function and also the hamiltonian depends only on the radial distance and there is no angular dependence anywhere.

02:36 So that's why i'm going to switch to spherical coordinates while carrying out these integrals and the spherical volume element is given by d cube x equal to r squared d r d omega where the old angular dependence is contained in the solid angle d omega so from zero to infinity we have r squared dr and from zero to four pi we have d omega now what is the square of this trial function.

03:13 We have e to the power minus 2 alpha r.

03:18 That's it.

03:21 So we have this constant tone.

03:24 As we said, there is no angular dependence anywhere.

03:27 So we can carry out this angular integration right away to obtain 4 pi.

03:33 Then we have this usual one -dimensional integration, dr, r squared, e to the power minus 2 alpha r.

03:44 I'm going to do a you substitution here i will let i will define u as 2 alpha r so that the u is equal to 2 alpha dr the integration limits will be the same because we are just rescaling our variable integration parameter there is no there is no fancy transformation here so we have 4 pi from 0 to infinity, du over 2 alpha, r squared u squared over 4 alpha squared over 4 alpha squared e to power minus u.

04:29 We have all these constant factors, we get 4 pi over 8 alpha cube times this integral du, u squared, u to power minus u.

04:47 Now if you are doing quantum mechanics, you should know this integral by heart essentially because you'll encounter it lots of times.

04:57 This is a definition of the factorial function indeed.

05:02 So the result of this integration is simply given by the factorial of the power of you here in front of the exponent.

05:10 So we get two factorial.

05:15 Okay.

05:17 Then the result of this calculation will be pi.

05:20 Over alpha cube.

05:24 Now let's do the expectation value of the hamiltonian, namely the numerator of the trial grand state energy.

05:34 So, si, h, si.

05:38 We have d -cube x, si -conjugate, h, h, the hamiltonian acting on the trial wave function.

05:48 Even though we have a real trial wayfunction, i'm going to keep this conjugate sign just for the same.

05:53 Of mathematical consistency.

05:57 Okay, we have d -cube x, si -star, minus 1 over 2 r squared.

06:09 Okay, i'm going to use the prime notation for the derivative, otherwise it will take so much space to write everything down.

06:19 So we have r squared times the derivative of the wave function, and we are taking another derivative.

06:27 Minus 1 over r times the wave function...

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