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3. Show by carrying out the appropriate integration that the total energy eigenfunctions for the harmonic...
A particle of mass m is bound by the spherically-symmetric three-dimensional harmonic- oscillator potential energy ,...
A particle of mass m is bound by the spherically-symmetric three-dimensional harmonic- oscillator potential energy , and ф are the usual spherical coordinates. (a) In the form given above, why is it clear that the potential energy function V) is (b) For this problem, it will be more convenient to express this spherically-symmetric where r , spherically symmetric? A brief answer is sufficient. potential energy in Cartesian coordinates x, y, and z as physically the same potential energy as the...
Consider the dimensionless harmonic oscillator Hamiltonian, (where m = h̄ = 1). Consider the orthogonal wave...
Consider the dimensionless harmonic oscillator Hamiltonian,
(where m = h̄ = 1).
Consider the orthogonal wave functions
and
, which are eigenfunctions of H with eigenvalues 1/2 and 5/2,
respectively.
with p=_ïda 2 2 We were unable to transcribe this imageY;(r) = (1-2x2)e-r2/2 (a) Let фо(x-AgVo(x) and φ2(x) = A2V2(x) and suppose that φ。(x) and φ2(x) are normalized. Find the constants Ao and A2. (b) Suppose that, at timet0, the state of the oscillator is given by Find the constant...
Consider a particle subjected to a harmonic oscillator potential of the form x)m. The allowed values...
Consider a particle subjected to a harmonic oscillator potential of the form x)m. The allowed values of energy for the simple harmonic oscillator is (a) What is the energy corresponding to the ground state (3 points)? (b) What is the energy separation between the ground state and the first excited state (3 points)? (c) The selection rule allows only those transitions for which the quantum number changes by 1. What is the energy of photon necessary to make the transition...
3. [Total: 24 pts] a) (8 pts) Calculate the probability of finding a particle in the...
3. [Total: 24 pts] a) (8 pts) Calculate the probability of finding a particle in the classically forbidden regime for the ground state of the 1D harmonic oscillator. Simplify the integral expression for the probability as much as possible - the integral can only be solved numerically. b) (8 pts) For the 1D harmonic oscillator, the energy eigenstates are either even or odd. This is indeed a special case of a more general statement: If V(x) is an even function...
First four harmonic oscillator normalized wavefunctions 1/4 Y.-(4)"-** 4, = 1/4 v2y ev2 1/4 Y, =|...
First four harmonic oscillator normalized wavefunctions 1/4 Y.-(4)"-** 4, = 1/4 v2y ev2 1/4 Y, =| -1)ev¾2 1/4 - 3y)e¬v³½ y =ax 1. Consider a harmonic oscillator with a = 1. a) Prove that these eigenstates are all orthonormal b) Plot the first four eigenstates. How would doubling the mass change the eigenfunctions? c) Pick one eigenstate, and show that it is a solution to the Schrodinger Equation, that is, show that V? on (x) + w²ma? ¢n (x) =...
Verify by direct integration that the functions are orthogonal with respect to the indicated weight function...
Verify by direct integration that the functions are orthogonal with respect to the indicated weight function w(x) on the given interval. 4p(x) = 1, 4,6x) = -x + 1, 12(X) = 2*2 - 2x - 2x + 1; w(x) - e*, [0, 0) Using integration by parts we find the following. (In the last step of each integral, simplify your answer completely.) 6 *wcx360(87%)/(x) dx = 60 1) ox 11-62 6o *wcxXq6x)22() dx = 6* 1) ox 11 + 1*2*x+...
Please solve for part (b) and (c) thank you! 1. Consider the function f(x) = e-x...
Please solve for part (b) and
(c) thank you!
1. Consider the function f(x) = e-x defined on the interval 0 < x < 1. (a) Give an odd and an even extension of this function onto the interval -1 < x < 1. Your answer can be in the form of an expression, or as a clearly labelled graph. [2 marks] (b) Obtain the Fourier sine and cosine representation for the functions found above. Hint: use integration by parts....
1 Particle in a Box with a Bump (based on B&J 4.11) Consider a particle of...
1 Particle in a Box with a Bump (based on B&J 4.11) Consider a particle of mass m in a 1-D double well with potential given by Vo, 05\x\<b V(x) = { 0, b<\x<c 100, [x]>c . We will study the lowest energy states, for which 0 <E<V, corresponding to tunnelling between the two wells. (a) Write down the time-independent Schödinger equation in the three regions -c<x<-b, –b< <b, and b< I< c. Write down the most general wavefunction solution...
Hi can you help me solve the following signal and system question on CTFS harmonic functions. Please show me how to get Cx[K] by integration by parts step by step. The graph for this question is not n...
Hi can you help me solve the following signal and system
question on CTFS harmonic functions. Please show me how to get
Cx[K] by integration by parts step by step. The graph for this
question is not needed. Thank you.
4. A periodic signal x(t) with a period of 4 seconds is described over one fundamental period by x(t) 3-t, 0<t<4. Graph the signal and find its CTFS harmonic function. Then graph on the same scale approximations to the signal...
ReviewI Constants TACTICS BOx 14.1 Identifying and analyzing simple harmonic motion Learning Goal: 1. If the...
ReviewI Constants TACTICS BOx 14.1 Identifying and analyzing simple harmonic motion Learning Goal: 1. If the net force acting on a particle is a linear restoring force, the motion will be simple harmonic motion around the equilibriunm To practice Tactics Box 14.1 Identifying and analyzing simple harmonic motion. position. 2. The position, velocity, and acceleration as a function of time are given in Synthesis 14.1 (Page 447) x(t)- Acos(2ft) Ug (t) = -(2rf)A sin( 2rft), A complete description of simple...
A particle of mass m is bound by the spherically-symmetric three-dimensional harmonic- oscillator potential energy , and ф are the usual spherical coordinates. (a) In the form given above, why is it clear that the potential energy function V) is (b) For this problem, it will be more convenient to express this spherically-symmetric where r , spherically symmetric? A brief answer is sufficient. potential energy in Cartesian coordinates x, y, and z as physically the same potential energy as the...
Consider the dimensionless harmonic oscillator Hamiltonian,
(where m = h̄ = 1).
Consider the orthogonal wave functions
and
, which are eigenfunctions of H with eigenvalues 1/2 and 5/2,
respectively.
with p=_ïda 2 2 We were unable to transcribe this imageY;(r) = (1-2x2)e-r2/2 (a) Let фо(x-AgVo(x) and φ2(x) = A2V2(x) and suppose that φ。(x) and φ2(x) are normalized. Find the constants Ao and A2. (b) Suppose that, at timet0, the state of the oscillator is given by Find the constant...
Consider a particle subjected to a harmonic oscillator potential of the form x)m. The allowed values of energy for the simple harmonic oscillator is (a) What is the energy corresponding to the ground state (3 points)? (b) What is the energy separation between the ground state and the first excited state (3 points)? (c) The selection rule allows only those transitions for which the quantum number changes by 1. What is the energy of photon necessary to make the transition...
3. [Total: 24 pts] a) (8 pts) Calculate the probability of finding a particle in the classically forbidden regime for the ground state of the 1D harmonic oscillator. Simplify the integral expression for the probability as much as possible - the integral can only be solved numerically. b) (8 pts) For the 1D harmonic oscillator, the energy eigenstates are either even or odd. This is indeed a special case of a more general statement: If V(x) is an even function...
First four harmonic oscillator normalized wavefunctions 1/4 Y.-(4)"-** 4, = 1/4 v2y ev2 1/4 Y, =| -1)ev¾2 1/4 - 3y)e¬v³½ y =ax 1. Consider a harmonic oscillator with a = 1. a) Prove that these eigenstates are all orthonormal b) Plot the first four eigenstates. How would doubling the mass change the eigenfunctions? c) Pick one eigenstate, and show that it is a solution to the Schrodinger Equation, that is, show that V? on (x) + w²ma? ¢n (x) =...
Verify by direct integration that the functions are orthogonal with respect to the indicated weight function w(x) on the given interval. 4p(x) = 1, 4,6x) = -x + 1, 12(X) = 2*2 - 2x - 2x + 1; w(x) - e*, [0, 0) Using integration by parts we find the following. (In the last step of each integral, simplify your answer completely.) 6 *wcx360(87%)/(x) dx = 60 1) ox 11-62 6o *wcxXq6x)22() dx = 6* 1) ox 11 + 1*2*x+...
Please solve for part (b) and
(c) thank you!
1. Consider the function f(x) = e-x defined on the interval 0 < x < 1. (a) Give an odd and an even extension of this function onto the interval -1 < x < 1. Your answer can be in the form of an expression, or as a clearly labelled graph. [2 marks] (b) Obtain the Fourier sine and cosine representation for the functions found above. Hint: use integration by parts....
1 Particle in a Box with a Bump (based on B&J 4.11) Consider a particle of mass m in a 1-D double well with potential given by Vo, 05\x\<b V(x) = { 0, b<\x<c 100, [x]>c . We will study the lowest energy states, for which 0 <E<V, corresponding to tunnelling between the two wells. (a) Write down the time-independent Schödinger equation in the three regions -c<x<-b, –b< <b, and b< I< c. Write down the most general wavefunction solution...
Hi can you help me solve the following signal and system
question on CTFS harmonic functions. Please show me how to get
Cx[K] by integration by parts step by step. The graph for this
question is not needed. Thank you.
4. A periodic signal x(t) with a period of 4 seconds is described over one fundamental period by x(t) 3-t, 0<t<4. Graph the signal and find its CTFS harmonic function. Then graph on the same scale approximations to the signal...
ReviewI Constants TACTICS BOx 14.1 Identifying and analyzing simple harmonic motion Learning Goal: 1. If the net force acting on a particle is a linear restoring force, the motion will be simple harmonic motion around the equilibriunm To practice Tactics Box 14.1 Identifying and analyzing simple harmonic motion. position. 2. The position, velocity, and acceleration as a function of time are given in Synthesis 14.1 (Page 447) x(t)- Acos(2ft) Ug (t) = -(2rf)A sin( 2rft), A complete description of simple...
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