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1. Show y = sin ax is not an eigenfunction of the operator d/dx, but is...
9. Show that the function w= sin(x) (n and a are constants) is an eigenfunction of...
9. Show that the function w= sin(x) (n and a are constants) is an eigenfunction of the Hamiltonian operator H = - raxz. What are the eigenvalues? hbar and m should be considered constant factors.
Show that the function (n and a are constants) is a eigenfunction of the Hamiltonian operator...
Show that the function (n and a are constants) is a eigenfunction of the Hamiltonian operator . What are the eigenvalues? and m should be considered constant factors. **SIDE NOTE: All the question marks should actually be upside down. I did not see the symbol for this! We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
qm 09.3 3. An operator  is Hermitian if it satisfies the condition $ $(y) dx...
qm 09.3
3. An operator  is Hermitian if it satisfies the condition $ $(y) dx = (Ap) u dx, for any wavefunctions $(x) and y(x). (i) The time dependent Schrödinger equation is ih au = fu, at where the Hamiltonian operator is Hermitian. Show that the equation of mo- tion for the expectation value of any Hermitian operator  is given by d(A) IH, Â]), dt ħi i = where the operator  does not depend explicitly on time....
2.The angular momentum is L = p a) What is the representation of the angular momentum operator b)Use the polar coordinates to compute L o)Show that the eigenfunction forp) m(p),where mis integer....
2.The angular momentum is L = p a) What is the representation of the angular momentum operator b)Use the polar coordinates to compute L o)Show that the eigenfunction forp) m(p),where mis integer. What is the Τηφ(p),where What mis integer. is the eigenfunction φ(p), assume 0 (p) 2π
2.The angular momentum is L = p a) What is the representation of the angular momentum operator b)Use the polar coordinates to compute L o)Show that the eigenfunction forp) m(p),where mis integer. What...
1. (50 points) Consider the particle in a one-dimensional box (0 s x S L). Assume a term is added to the Hamiltonian of the form: πχ V(x)g sin Sketch the potential and the expected eigenfunction (...
1. (50 points) Consider the particle in a one-dimensional box (0 s x S L). Assume a term is added to the Hamiltonian of the form: πχ V(x)g sin Sketch the potential and the expected eigenfunction (small g). In the limit of small g, find the second order correction to the ground state energy 2. (50 points) For a diatomic molecule rotating in free space, the Hamiltonian may be written: 12 21 Where L is the total angular momentum operator,...
(a) Show that (@) = sin e- is an eigenfunction of both Î, and Î", where...
(a) Show that (@) = sin e- is an eigenfunction of both Î, and Î", where = -1 1 a 1 22 sin + sin 020 sin0 and derive the corresponding eigenvalues. You may use the identity 1 a 1 sin sin 2 sin sin 0 80 sino 31 (sin 00 (5 marks) (6) Consider the function $(,0,4)= A - 1/200 sin 6e-ip, 20 where A is a constant and an is the Bohr radius. This is a hydrogen atom...
1) (35 points) The wave function for a particle moving along x axis between the limits 0 and L is: (x)-C sin (nx xL) where n are 1, 2, 3, A) Determine the normalization constant C B) Why can'...
1) (35 points) The wave function for a particle moving along x axis between the limits 0 and L is: (x)-C sin (nx xL) where n are 1, 2, 3, A) Determine the normalization constant C B) Why can't n take the value of 0, briefly explain C) For n-3 determine the values of x (in terms of L) that correspond to a maximum or a minimum in the wave function D) For n-3 determine the values of x (in...
Consider the following initial value problem: dy = sin(x - y) dx, y(0) 1. Write the...
Consider the following initial value problem: dy = sin(x - y) dx, y(0) 1. Write the equation in the form ay = G(ax +by+c), dx where a, b, and c are constants and G is a function. 2. Use the substitution u = ax + by + c to transfer the equation into the variables u and x only. 3. Solve the equation in (2). 4. Re-substitute u = ax + by + c to write your solution in terms...
Exercise 1: The helium atom and spin operators 26 pts (a) Show that the expectation value...
Exercise 1: The helium atom and spin operators 26 pts (a) Show that the expectation value of the Hamiltonian in the (sa)'(2a)' excited state of helium is given by E = $42.0) (Avo ) anordes ++f63,(-) (%13-12 r) 62(e)drz + løn.(r.) per 142, (ra)]" drų dr2 - / 01.(ru) . (ra) Anemia 02.(r.)61.(r.)dr; dr2 (1) Use the approximate, antisymmetrized triplet state wave function for the (Isa)'(280)' state as discussed in class. Hint: make use of the orthonormality of the hydrogenic...
1. (25 points) The Hamiltonian operator H, for a particular molecule has a complete set of...
1. (25 points) The Hamiltonian operator H, for a particular molecule has a complete set of orthonormal eigenfunctions on (where n = 1,2,3,...) with corresponding eigenvalues (n-1)h. The molecule in state n is subject to measurement of the dipole moment, for which the mathematical operator is represented by M. After the measurement of the dipole moment, the wave function of the particle is: Y = 0.5 0.1 +0.7071 Q. + 0.5 Pn+1 a) Show that is normalized. Now consider a...
9. Show that the function w= sin(x) (n and a are constants) is an eigenfunction of the Hamiltonian operator H = - raxz. What are the eigenvalues? hbar and m should be considered constant factors.
qm 09.3
3. An operator  is Hermitian if it satisfies the condition $ $(y) dx = (Ap) u dx, for any wavefunctions $(x) and y(x). (i) The time dependent Schrödinger equation is ih au = fu, at where the Hamiltonian operator is Hermitian. Show that the equation of mo- tion for the expectation value of any Hermitian operator  is given by d(A) IH, Â]), dt ħi i = where the operator  does not depend explicitly on time....
2.The angular momentum is L = p a) What is the representation of the angular momentum operator b)Use the polar coordinates to compute L o)Show that the eigenfunction forp) m(p),where mis integer. What is the Τηφ(p),where What mis integer. is the eigenfunction φ(p), assume 0 (p) 2π
2.The angular momentum is L = p a) What is the representation of the angular momentum operator b)Use the polar coordinates to compute L o)Show that the eigenfunction forp) m(p),where mis integer. What...
1. (50 points) Consider the particle in a one-dimensional box (0 s x S L). Assume a term is added to the Hamiltonian of the form: πχ V(x)g sin Sketch the potential and the expected eigenfunction (small g). In the limit of small g, find the second order correction to the ground state energy 2. (50 points) For a diatomic molecule rotating in free space, the Hamiltonian may be written: 12 21 Where L is the total angular momentum operator,...
(a) Show that (@) = sin e- is an eigenfunction of both Î, and Î", where = -1 1 a 1 22 sin + sin 020 sin0 and derive the corresponding eigenvalues. You may use the identity 1 a 1 sin sin 2 sin sin 0 80 sino 31 (sin 00 (5 marks) (6) Consider the function $(,0,4)= A - 1/200 sin 6e-ip, 20 where A is a constant and an is the Bohr radius. This is a hydrogen atom...
1) (35 points) The wave function for a particle moving along x axis between the limits 0 and L is: (x)-C sin (nx xL) where n are 1, 2, 3, A) Determine the normalization constant C B) Why can't n take the value of 0, briefly explain C) For n-3 determine the values of x (in terms of L) that correspond to a maximum or a minimum in the wave function D) For n-3 determine the values of x (in...
Consider the following initial value problem: dy = sin(x - y) dx, y(0) 1. Write the equation in the form ay = G(ax +by+c), dx where a, b, and c are constants and G is a function. 2. Use the substitution u = ax + by + c to transfer the equation into the variables u and x only. 3. Solve the equation in (2). 4. Re-substitute u = ax + by + c to write your solution in terms...
Exercise 1: The helium atom and spin operators 26 pts (a) Show that the expectation value of the Hamiltonian in the (sa)'(2a)' excited state of helium is given by E = $42.0) (Avo ) anordes ++f63,(-) (%13-12 r) 62(e)drz + løn.(r.) per 142, (ra)]" drų dr2 - / 01.(ru) . (ra) Anemia 02.(r.)61.(r.)dr; dr2 (1) Use the approximate, antisymmetrized triplet state wave function for the (Isa)'(280)' state as discussed in class. Hint: make use of the orthonormality of the hydrogenic...
1. (25 points) The Hamiltonian operator H, for a particular molecule has a complete set of orthonormal eigenfunctions on (where n = 1,2,3,...) with corresponding eigenvalues (n-1)h. The molecule in state n is subject to measurement of the dipole moment, for which the mathematical operator is represented by M. After the measurement of the dipole moment, the wave function of the particle is: Y = 0.5 0.1 +0.7071 Q. + 0.5 Pn+1 a) Show that is normalized. Now consider a...
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