The wave functions of the H-atom for the ground state and the first excited state are given by Yo...
The wave functions of the H-atom for the ground state and the first excited state are given by Yoo(θ, φ), 100(r' exp - , 200(r, θ, φ) a. Show that these 5 wave-functions are all mutually orthogonal to each other. b. Determine the expectation values(r2〉nl of the operator r2 defined as follows 〈r2〉10-rd3TV1,00(a) r2ψ1,0,0(2) and 〈r2〉20, 〈r2)21 defined analogously
Problem 5 (a) Two (unnormalized) excited state wavefunctions of the H atom are (1) 4 = (2-)e-r/ao (ii) W = r sinô coso e-r/220 Normalize both functions to be 1. (b) Confirm that these two functions are mutually orthogonal
( 25 marks) The wave function for a hydrogen atom in the ground state is given by \(\psi(r)=A e^{-r / a_{s}}\), where \(A\) is a constant and \(a_{B}\) is the Bohr radius. (a) Find the constant \(A\). (b) Determine the expectation value of the potential energy for the ground state of hydrogen.
[12%] The ground state wave function for hydrogen atom is (a) N exp(-r/u2) (c) Nr2 exp(-r2/a ) (d) Nexp㈠,21%) (e) N exp(-rlao), where N is the normalized factor and ao is the Bohr radius.
Consider the excited state wave function for He atom given by the following Slater determinant 1 432,0(1) V3.2,-2B(1) He (1,2)= V2 V3.2,a(2) W32,-2B(2) Here Y 3,2,-and Y3,2,-2 are hydrogenic wave functions (with Z = 2, see the equation sheet). Show that He (1, 2) is an eigenfunction of Î. = Î., +Î.2. What is the eigenvalue? Î.,, ..2, and Î, are the z-components of the orbital angular momentum operators for electrons 1 and 2, and the z-component of the total...
2. The hydrogen atom [8 marks] The time-independent Schrödinger equation for the hydrogen atom in the spherical coordinate representation is where ao-top- 0.5298 10-10rn is the Bohr radius, and μ is the electon-proton reduced mass. Here, the square of the angular momentum operator L2 in the spherical coordinate representation is given by: 2 (2.2) sin θー sin θ 00 The form of the Schrödinger equation means that all energy eigenstates separate into radial and angular motion, and we can write...
The ground state wave function of the hydrogen atom is given by 1 (r) = 7/a. Vπα3 What is the ground state wave function of the hydrogen atom in momentum space? Hint: Choose the z-axis along the momentum direction.
1. Given a state y(r) expanded on the eigenstates of the Hamiltonian for the electron, H, in a hydrogen atom: where the subscript of E is n, the principal quantum number. The other two numbers are the 1 and m values, find the expectation values of H (you may use the eigenvalue equation to evaluate for H), L-(total angular momentum operator square), Lz (the z-component of the angular momentum operator) and P (parity operator). Draw schematic pictures of 1 and...
ANSWER ALL QUESTIONS 1. (a) The hydrogen atom wave functions are written as Unim. State the values of n, I, and m. State the relation between a physical quantity and each quantum number. At =0 the hydrogen atom is in the superposition state (7,0) = 4200 + A¥210 V3 where A is a real positive constant. Find A by normalization and determine the wave function at time t > 0. Find the average energy of the electron in eV given...
ANSWER ALL QUESTIONS 1. (a) The hydrogen atom wave functions are written as Unim. State the values of n, I, and m. State the relation between a physical quantity and each quantum number. At =0 the hydrogen atom is in the superposition state (7,0) = 4200 + A¥210 V3 where A is a real positive constant. Find A by normalization and determine the wave function at time t > 0. Find the average energy of the electron in eV given...