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2. Two noninteracting particles, each of mass m, are in the 1-D harmonic oscillator potential Describe...
Problem 5.7 Two noninteracting particles (of equal mass) share the same harmonic oscillator potential, one in...
Problem 5.7 Two noninteracting particles (of equal mass) share the same harmonic oscillator potential, one in the ground state and one in the first excited state. (a) Construct the wave function, y (x1, x2), assuming (1) they are distinguishable, (ii) they are identical bosons, (iii) they are identical fermions. Plot IV (x1,x2)|in each case (use, for instance, Mathematica's Plot3D). (6) Use Equations 5.23 and 5.25 to determine ((x1 - x2) for each case. (C) Express each y (x1, x2) in...
2096) Two noninteracting particles 1 and 2, each of mass m, are in a 1-D infinite...
2096) Two noninteracting particles 1 and 2, each of mass m, are in a 1-D infinite square well ol width a. If one is in the state V'in and the other in the state (n! /), calculate C(xI-x), assuming (a) (6%) they are distinguishable particles, (b) (7%) Ihey are identical bosons, and (c) (796) Ihey arc identical fermions. 4.
Q1: Consider two particles occupying the ground state 01) and the first excited state (42) of...
Q1: Consider two particles occupying the ground state 01) and the first excited state (42) of the one dimensional infinite square potential well. Give a proper form of the normalized wave function of the system (11,12) in the following cases: (a) The two particles are distinguishable. (b) The two particles are identical bosons. (c) The two particles are identical fermions
6. (Extra Credit: 6 Points) Consider two noninteracting particles of mass m in an infinite square well of width L. For the case with one particle in the single-particle state In) and the other in...
6. (Extra Credit: 6 Points) Consider two noninteracting particles of mass m in an infinite square well of width L. For the case with one particle in the single-particle state In) and the other in the state k) (nメk), calculate the expectation value of the squared inter-particle spacing (71-72) , assuming (a) the particles are distinguishable, (b) the particles are identical in a symmetrical spatial state, and (c) the particles are identical in an anti-symmetric spatial state. Use Dirac notation...
tthe-independent Help: The operator expression dimensions is given by H 2m r ar2 [2] A particle of mass m is in a three-dimensional, spherically symmetric harmonic oscillator potential given by V...
tthe-independent Help: The operator expression dimensions is given by H 2m r ar2 [2] A particle of mass m is in a three-dimensional, spherically symmetric harmonic oscillator potential given by V(r)2r2. The particle is in the I-0 state. Noting that all eigenfunetions must be finite everywhere, find the ground-state radial wave-function R() and the ground-state energy. You do not have to nor oscillator is g (x) = C x exp(-8x2), where C and B are constants) harmonic malize the solution....
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...
[1 44= 9 marks ] Question 5 Consider two identical particles in 1D which exist in single-particle (normalised) (x), and...
[1 44= 9 marks ] Question 5 Consider two identical particles in 1D which exist in single-particle (normalised) (x), and are in such close proximity they can be considered as indistinguishable. wave functions /a(x) and (a) Write down the symmetrised two-particle wave function for the case where the particles are bosons (VB) and the case where the particles are fermions (Vp). (b) Show that the expectation value (xjr2)B,F is given by: (T122) в,F — (а:)a (х)ь + dx x y:(")...
1. Imagine that a particle can exist in one of the three states: |0 > , |1 >, |2> . We now consider 2 such part...
1. Imagine that a particle can exist in one of the three states: |0 > , |1 >, |2> . We now consider 2 such particles. How many distinguishable states are possible if the particles are (a) distinguishable (b) indistinguishable, classic (c) bosons (d) fermions. Write down the wave function of the system for all cases.
1. Position representation of the harmonic oscillator wave functions. (a) Using that the position representation of...
1. Position representation of the harmonic oscillator wave functions. (a) Using that the position representation of the ground state of the harmonic ) _ (쁩)1/4e-mura/an, find 너 1) and (212) (2 points) (b) Verify explicitly that your solution for (r|1) fulfills the position representa- oscillator is (rlo tion of the Schrödinger equation (1 point) (1 point) (c) What are the corresponding energy eigenvalues En?
A particle of charge q and mass m is bound in the ground state of a one-dimensional harmonic osci...
A particle of charge q and mass m is bound in the ground state of a one-dimensional harmonic oscillator potential with frequency oo. At time t-0 a weak spatially uniform electric field (E) is turned on, so that the perturbation to the Hamiltonian can be described as R'(t) =-q Exe-t/t for t> 0. Using first order, time-dependent perturbation theory, calculate the following probabilities: (a) the particle is detected in the first excited state after a very long time (t »...
Problem 5.7 Two noninteracting particles (of equal mass) share the same harmonic oscillator potential, one in the ground state and one in the first excited state. (a) Construct the wave function, y (x1, x2), assuming (1) they are distinguishable, (ii) they are identical bosons, (iii) they are identical fermions. Plot IV (x1,x2)|in each case (use, for instance, Mathematica's Plot3D). (6) Use Equations 5.23 and 5.25 to determine ((x1 - x2) for each case. (C) Express each y (x1, x2) in...
2096) Two noninteracting particles 1 and 2, each of mass m, are in a 1-D infinite square well ol width a. If one is in the state V'in and the other in the state (n! /), calculate C(xI-x), assuming (a) (6%) they are distinguishable particles, (b) (7%) Ihey are identical bosons, and (c) (796) Ihey arc identical fermions. 4.
Q1: Consider two particles occupying the ground state 01) and the first excited state (42) of the one dimensional infinite square potential well. Give a proper form of the normalized wave function of the system (11,12) in the following cases: (a) The two particles are distinguishable. (b) The two particles are identical bosons. (c) The two particles are identical fermions
6. (Extra Credit: 6 Points) Consider two noninteracting particles of mass m in an infinite square well of width L. For the case with one particle in the single-particle state In) and the other in the state k) (nメk), calculate the expectation value of the squared inter-particle spacing (71-72) , assuming (a) the particles are distinguishable, (b) the particles are identical in a symmetrical spatial state, and (c) the particles are identical in an anti-symmetric spatial state. Use Dirac notation...
tthe-independent Help: The operator expression dimensions is given by H 2m r ar2 [2] A particle of mass m is in a three-dimensional, spherically symmetric harmonic oscillator potential given by V(r)2r2. The particle is in the I-0 state. Noting that all eigenfunetions must be finite everywhere, find the ground-state radial wave-function R() and the ground-state energy. You do not have to nor oscillator is g (x) = C x exp(-8x2), where C and B are constants) harmonic malize the solution....
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...
[1 44= 9 marks ] Question 5 Consider two identical particles in 1D which exist in single-particle (normalised) (x), and are in such close proximity they can be considered as indistinguishable. wave functions /a(x) and (a) Write down the symmetrised two-particle wave function for the case where the particles are bosons (VB) and the case where the particles are fermions (Vp). (b) Show that the expectation value (xjr2)B,F is given by: (T122) в,F — (а:)a (х)ь + dx x y:(")...
1. Position representation of the harmonic oscillator wave functions. (a) Using that the position representation of the ground state of the harmonic ) _ (쁩)1/4e-mura/an, find 너 1) and (212) (2 points) (b) Verify explicitly that your solution for (r|1) fulfills the position representa- oscillator is (rlo tion of the Schrödinger equation (1 point) (1 point) (c) What are the corresponding energy eigenvalues En?
A particle of charge q and mass m is bound in the ground state of a one-dimensional harmonic oscillator potential with frequency oo. At time t-0 a weak spatially uniform electric field (E) is turned on, so that the perturbation to the Hamiltonian can be described as R'(t) =-q Exe-t/t for t> 0. Using first order, time-dependent perturbation theory, calculate the following probabilities: (a) the particle is detected in the first excited state after a very long time (t »...
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