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2. (8 points) Consider an electron in a hydrogen atom with 1. There are three possible...
1. Given a state y(r) expanded on the eigenstates of the Hamiltonian for the electron, H, in a hydrogen atom: where the...
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...
2. The hydrogen atom [8 marks] The time-independent Schrödinger equation for the hydrogen atom in...
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...
HW12.2 (3 points) [1,1],[1,0),[1,-1) are the three eigenstates of Lạ& Ly with quantum number l =...
HW12.2 (3 points) [1,1],[1,0),[1,-1) are the three eigenstates of Lạ& Ly with quantum number l = 1 and m = 1,0,-1, respectively. a) (1 point) What are the following states Lx|1,1) =? Lx|1,0) = ? Lx|1, -1) = ? b) (1 point) Find out eigenstates of Lx that are linear combinations of (1,1),(1,0), and (1,-1) What is the corresponding eigenvalue for each of the eigenstate you find? c) (1 point) What is the matrix representation of Lx in the basis...
Exercise 2:Commutators Given (AB, C) - ABC - ACB + ACB-CAB - ABC] + [A, CJB....
Exercise 2:Commutators Given (AB, C) - ABC - ACB + ACB-CAB - ABC] + [A, CJB. 1- Show that the commutator[L.L]is equal to zero. 2-Computethecommutator(Ly, Lx]. 3-Compute the commutator 3.Lx]. 4-Compute the commutator[LLY.Lx]. 5- Add the results from (2)-(4) to compute [LLyLx]. Lxercise 3: Matrix elements The angular momentum operators acting on the angular momentum eigenstates. Il determined by L-11.m) = (1 + 1) - m(m + 1)2.m +1) L_11.m) = /(1+1) - mm - 1)/1.m-1) L211,m) = hm|1,m) 1-...
Problem 5. (30 points) Consider a Harmonic oscillator with H that H=(ata + 1 / 2)ho,...
Problem 5. (30 points) Consider a Harmonic oscillator with H that H=(ata + 1 / 2)ho, where a=dma)X + i (a) (4 points) Show P, and a x 2h 2h 2moh P. Show also 2moh that [a, a]-l. (b) (6 points) Starting from the commuters la, HJand la', A), where H-H(h) show that the eigenvalues of Hare e,=(n+1/2) for n-0, 1,2, Show also that alm)-nln-l), and a l). (( points) Find the normalized ground state wavefunction by projecting alo)-0 on...
SG 3 *** 11- SG 2 Y- SG 1 Trap Trap Figure 1: Schematic illustration of...
SG 3 *** 11- SG 2 Y- SG 1 Trap Trap Figure 1: Schematic illustration of a the stream of atoms described, each individually pre- pared in the superposition V = 1 + , being measured by sequential Stern-Gerlach machines. 2. (20 points) Spin of an electron is a quantum mechanical observable, and as such, there are corresponding operators for electron spin. For example, the operator S, relates to the observable called 2-spin' and has exactly two associated eigenfunctions and...
1 L, as a dynamical system (Notes from Assignment #2) We take our definition of dynamical system ...
1 L, as a dynamical system (Notes from Assignment #2) We take our definition of dynamical system to be an "object" along with a specific set of modifications that can be performed (dynamically) upon this object. In this case, the object is a bi-infinite straight road with a lamp post at every street corner and a marked lamp (the position of the lamplighter). There are two possible types of modifications: the lamplighter can walk any distance in either direction from...
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...
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...
HW12.2 (3 points) [1,1],[1,0),[1,-1) are the three eigenstates of Lạ& Ly with quantum number l = 1 and m = 1,0,-1, respectively. a) (1 point) What are the following states Lx|1,1) =? Lx|1,0) = ? Lx|1, -1) = ? b) (1 point) Find out eigenstates of Lx that are linear combinations of (1,1),(1,0), and (1,-1) What is the corresponding eigenvalue for each of the eigenstate you find? c) (1 point) What is the matrix representation of Lx in the basis...
Exercise 2:Commutators Given (AB, C) - ABC - ACB + ACB-CAB - ABC] + [A, CJB. 1- Show that the commutator[L.L]is equal to zero. 2-Computethecommutator(Ly, Lx]. 3-Compute the commutator 3.Lx]. 4-Compute the commutator[LLY.Lx]. 5- Add the results from (2)-(4) to compute [LLyLx]. Lxercise 3: Matrix elements The angular momentum operators acting on the angular momentum eigenstates. Il determined by L-11.m) = (1 + 1) - m(m + 1)2.m +1) L_11.m) = /(1+1) - mm - 1)/1.m-1) L211,m) = hm|1,m) 1-...
Problem 5. (30 points) Consider a Harmonic oscillator with H that H=(ata + 1 / 2)ho, where a=dma)X + i (a) (4 points) Show P, and a x 2h 2h 2moh P. Show also 2moh that [a, a]-l. (b) (6 points) Starting from the commuters la, HJand la', A), where H-H(h) show that the eigenvalues of Hare e,=(n+1/2) for n-0, 1,2, Show also that alm)-nln-l), and a l). (( points) Find the normalized ground state wavefunction by projecting alo)-0 on...
SG 3 *** 11- SG 2 Y- SG 1 Trap Trap Figure 1: Schematic illustration of a the stream of atoms described, each individually pre- pared in the superposition V = 1 + , being measured by sequential Stern-Gerlach machines. 2. (20 points) Spin of an electron is a quantum mechanical observable, and as such, there are corresponding operators for electron spin. For example, the operator S, relates to the observable called 2-spin' and has exactly two associated eigenfunctions and...
1 L, as a dynamical system (Notes from Assignment #2) We take our definition of dynamical system to be an "object" along with a specific set of modifications that can be performed (dynamically) upon this object. In this case, the object is a bi-infinite straight road with a lamp post at every street corner and a marked lamp (the position of the lamplighter). There are two possible types of modifications: the lamplighter can walk any distance in either direction from...
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