Question 1 (413' points) What are the tneohonieai properties of a i'iydrogeh atom? To answer this question we need to introduce some new concepts. First. we need to evaluate how the energy of the electron changes when we change the radius of the wave function. Here. we are going to use an approximation technique for the evaluation of the ground state wave function called the variational principle: the expectation vahte of the Hamiltonian operator (i.e., the expectation value of the energy is minimum for the exact ground state of the problem. This approximation technique allows us to try different guesses for the shape of the ground-state wave function that contain adjustable parameters. and then tune the value of those parameters until the expectation value of the energy is minimized. Here we are going to cheat and use a shape of the radial wave function that is a decaying exponential. i.e.. that we loiow is the solution to Schrodinger's equation. but we will pretend that we do not know the coefficient of the exponential function {i.e._. how localized the ground-state wave function is). Let's call the tullenown coefficient X: 2 , _ .Jr X R15(T.X) X:V2t. \"I; {a} Use the equation for R;;[r..1'} above to show that the expectation value of the Hamiltonian operator (in spherical coordinates}. i.e. the expectation value for the total energy {kinetic plus potential} of the state EIIX) as a function of the parameter I (i.e.. the parameter that determines the shape of the wave function) is given by: a h E X = . t J X. X. T12 e2 where the constants are a = and h = . 2m three {b} Show that the expectation value is indeed minimum for X=a-:]. as you expected. (c) How. let=s evaluate the bulk modulus of the H atom. The bulk modulus quantifies how much the fractional volume changes (til/TV) by an incremental change to the pressure trip. This can be combined with the equation for the pressure P = (iE.-"(iV. Use the result in part {a} to calculate the bulk modulus according to: a _ v as _ Vegan!) _ 3 W where I" is the volume of the atom. and the second derivative is evaluated at the minimum of the energy-va-x-'olume curve. {d} lCompare the bulk modulus you found in part (c) with the bulk modulus of aluminum. steel. and diamond. What does the result of this exercise tell you about the mechanical properties of materials?I Hints . The wave function is given by (in Spherical coordinater ) 4 ( r, 0. 0 ) = Ris ( r) y (o,8 ) = 2 - r/x and is spherically symmetric . The Hamiltonian operator in spherical coordinater in H = _ t 2 2 m + V (r ) => kinetic energy potential every Ay 2 ar + 13mozo ( smo 24 ) 29 / r'sin 29 2de + K. E. + 2 e r 4 P. E. 20P. E. 20 . You need to evaluate E ( X ) = * ( 1 0 4 ) Hulroe ) rsing 2 1=0 9= 0 6= 0 drdody = = + . What are the derivativer with respect to the angler I and s of afunction that depends only on r? . You only need to evaluate the following two integrals. Sore 2r/ x dr and 2 - 2 /x dr re . The volume of the hydrogen atom is assumed to be V = #n X' Convert E (X) to E(v)