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Consider a wave function that is an eigenfunction of L2 with the eigenvalue 42h2. What are...
Consider the excited state wave function for He atom given by the following Slater determinant 1...
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...
A function Ψ(x) is an eigenfunction of an operator A with an eigenvalue λ if Ay(x)-AW(x)...
A function Ψ(x) is an eigenfunction of an operator A with an eigenvalue λ if Ay(x)-AW(x) where λ is some number. Show that the function ψ(x)-xe-rn is an eigenfunction of the operator A--x2. What is the eigenvalue?
1. Show y = sin ax is not an eigenfunction of the operator d/dx, but is...
1. Show y = sin ax is not an eigenfunction of the operator d/dx, but is an eigenfunction of the operator da/dx. 2. Show that the function 0 = Aeimo , where i, m, and A are constants, is an eigenfunction of the angular momentum operator is the z-direction: M =; 2i ap' and what are the eigenvalues? 3. Show the the function y = Jź sin MA where n and L are constants, is an eigenfunction of the Hamiltonian...
5. Part 1. (6 pt) An electron moves around a 2D ring with ring radius 0.50 nm in the state m --20...
5. Part 1. (6 pt) An electron moves around a 2D ring with ring radius 0.50 nm in the state m --20. Determine the wavelength (in nm) of the particle wave induced by this electron. (similar to a question in Exam 1) Part 2. (a) (7pt) A wavefunction is given by y, (e, 4-B sin cos(6). Can this function be an eigenfunction of Legendrían operator (A2.sunagatsineaesin暘for a quantum particle moving around a spherical surface)? If so, determine the eigenvalue and...
Spin Wave Function For a system consisting of two electrons, Show that either tye symmetric func...
Spin Wave Function
For a system consisting of two electrons,
Show that either tye symmetric function or the antisymmetric
function is an eigenfunction of the total spin operator ,S^2
What is the eigenvalue for the wave function you chose?
For a system consisting of two electrons, show that either the symmetric function ψ.-1(d)β(2) + a(2g(1)] or the antisymmetric function: is an eigenfunction of the total spin operator, 32 What is the eigenvalue for the wave function you chose?
For a...
Consider an electron whose wave function is ?(r,0,?)-- e* sin ? + cos ?)f(r). 47t where...
Consider an electron whose wave function is ?(r,0,?)-- e* sin ? + cos ?)f(r). 47t where I (rrr2dr-| , and ?, ? are the azimuth and polar angles, respectively. (i) Rewrite the wave function in terms of the appropriate spherical harmonics. (4 marks) (ii) What are the possible measurement results of the z-component L, of the angular momentum of the electron in this state? (6 marks) (iii) Calculate the probability of obtaining each of the possible results in part (i)....
Problem 4. Imagine you are a biophysical chemist and are interested in electron transport by membrane...
Problem 4. Imagine you are a biophysical chemist and are interested in electron transport by membrane proteins. Let us consider the following wave function: (The above wave function corresponds to a free particle for which the potential energy V= 0) (a) Is the above wave function an eigenfunction of the momentum operator p? (b) Is the above wave function an eigenfunction of the operator for the momentum squared p2 ? (c) If your answer to part (b) is yes, then...
322 CHAPTER 5. ANGULAR MOMENTUM Problem 5.12 Consider a particle whose wave function is 1 222-x2-y2...
322 CHAPTER 5. ANGULAR MOMENTUM Problem 5.12 Consider a particle whose wave function is 1 222-x2-y2 4 A 3 xz (x, y, z) = 2 2 Calculate L2 (x, y, z) and L-y(x, y, z). Find the total angular momentum of this particle. (b) Calculate L+ y (x, y, z) and (Y L+ W). (c)If a measurement of the z-component of the orbital angular momentum is carried out, find the probabilities corresponding to finding the results 0, h, and -h....
Consider the following wave function: y(x, t) = cos(kx - omega t). a. Show that the...
Consider the following wave function: y(x, t) = cos(kx - omega t). a. Show that the above function is an eigenfunction of the operator partialdifferential^2/partialdifferential x^2[...] and determine its eigenvalue. b. Show that the above function is a solution of the wave equation expressed as partialdifferential^2 y(x, t)/partialdifferential x^2 = 1/v^2 partialdifferential^2 y(x, t)/partialdifferential t^2, given the wave velocity is v = omega/k (where omega = 2 pi V and k = 2pi/lambda).
3. Consider a rigid rotor whose Hamiltonian is given by H L2(21) where L is the angular momentum operator and I is the...
3. Consider a rigid rotor whose Hamiltonian is given by H L2(21) where L is the angular momentum operator and I is the moment of inertia of the rotator. Its rotation is described by a wave function: (0, N{Yo0(0,6)(1 3i) Y1-1(0,6) 2 Y21(0.0) Y20(0.) Find the normalization constant, N. (i) Find the probability to occupy state Yo0- (ii Find the expectation value of L2 of this state (iii Find the expectation value of L2 of this state (iv) Find (L2L2/21...
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...
A function Ψ(x) is an eigenfunction of an operator A with an eigenvalue λ if Ay(x)-AW(x) where λ is some number. Show that the function ψ(x)-xe-rn is an eigenfunction of the operator A--x2. What is the eigenvalue?
1. Show y = sin ax is not an eigenfunction of the operator d/dx, but is an eigenfunction of the operator da/dx. 2. Show that the function 0 = Aeimo , where i, m, and A are constants, is an eigenfunction of the angular momentum operator is the z-direction: M =; 2i ap' and what are the eigenvalues? 3. Show the the function y = Jź sin MA where n and L are constants, is an eigenfunction of the Hamiltonian...
5. Part 1. (6 pt) An electron moves around a 2D ring with ring radius 0.50 nm in the state m --20. Determine the wavelength (in nm) of the particle wave induced by this electron. (similar to a question in Exam 1) Part 2. (a) (7pt) A wavefunction is given by y, (e, 4-B sin cos(6). Can this function be an eigenfunction of Legendrían operator (A2.sunagatsineaesin暘for a quantum particle moving around a spherical surface)? If so, determine the eigenvalue and...
Spin Wave Function
For a system consisting of two electrons,
Show that either tye symmetric function or the antisymmetric
function is an eigenfunction of the total spin operator ,S^2
What is the eigenvalue for the wave function you chose?
For a system consisting of two electrons, show that either the symmetric function ψ.-1(d)β(2) + a(2g(1)] or the antisymmetric function: is an eigenfunction of the total spin operator, 32 What is the eigenvalue for the wave function you chose?
For a...
Consider an electron whose wave function is ?(r,0,?)-- e* sin ? + cos ?)f(r). 47t where I (rrr2dr-| , and ?, ? are the azimuth and polar angles, respectively. (i) Rewrite the wave function in terms of the appropriate spherical harmonics. (4 marks) (ii) What are the possible measurement results of the z-component L, of the angular momentum of the electron in this state? (6 marks) (iii) Calculate the probability of obtaining each of the possible results in part (i)....
Problem 4. Imagine you are a biophysical chemist and are interested in electron transport by membrane proteins. Let us consider the following wave function: (The above wave function corresponds to a free particle for which the potential energy V= 0) (a) Is the above wave function an eigenfunction of the momentum operator p? (b) Is the above wave function an eigenfunction of the operator for the momentum squared p2 ? (c) If your answer to part (b) is yes, then...
322 CHAPTER 5. ANGULAR MOMENTUM Problem 5.12 Consider a particle whose wave function is 1 222-x2-y2 4 A 3 xz (x, y, z) = 2 2 Calculate L2 (x, y, z) and L-y(x, y, z). Find the total angular momentum of this particle. (b) Calculate L+ y (x, y, z) and (Y L+ W). (c)If a measurement of the z-component of the orbital angular momentum is carried out, find the probabilities corresponding to finding the results 0, h, and -h....
Consider the following wave function: y(x, t) = cos(kx - omega t). a. Show that the above function is an eigenfunction of the operator partialdifferential^2/partialdifferential x^2[...] and determine its eigenvalue. b. Show that the above function is a solution of the wave equation expressed as partialdifferential^2 y(x, t)/partialdifferential x^2 = 1/v^2 partialdifferential^2 y(x, t)/partialdifferential t^2, given the wave velocity is v = omega/k (where omega = 2 pi V and k = 2pi/lambda).
3. Consider a rigid rotor whose Hamiltonian is given by H L2(21) where L is the angular momentum operator and I is the moment of inertia of the rotator. Its rotation is described by a wave function: (0, N{Yo0(0,6)(1 3i) Y1-1(0,6) 2 Y21(0.0) Y20(0.) Find the normalization constant, N. (i) Find the probability to occupy state Yo0- (ii Find the expectation value of L2 of this state (iii Find the expectation value of L2 of this state (iv) Find (L2L2/21...
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