Homework Answers
a) If the particles are distinguishable, then let the first particle be in the ground state and the second particle be in the first excited state. This state is represented as
Another possibility is the first particle be in first excited state and the second particle is in the ground state. This state is represented as
b) If the particles are identical then either particle can be in ground state or first excited state. For bosons, the wave function is symmetric and the normalization factor is 1/sqrt(2).
c) For identical fermions, the wave function is anti-symmetric and the normalization factor is 1/sqrt(2).
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Q1: Consider two particles occupying the ground state 01) and the first excited state (42) of...
2. Two noninteracting particles, each of mass m, are in the 1-D harmonic oscillator potential Describe...
2. Two noninteracting particles, each of mass m, are in the 1-D harmonic oscillator potential Describe their ground state and the next two excited states, that is, their corresponding (i) wave functions, (i) energy eigenvalues, and (ii) degeneracies if two particles are (a) distinguishable particles, (b) the identical bosons, and (c) the identical fermions. 3. Suppose one particle is in the ground state, and the other is in the first excited state for tw particles described in Prob. 2. Calculate...
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.
Quantum Mechanics. Find the energies, degenerations and wave functions for the first three energy levels (ground state...
Quantum Mechanics. Find the energies, degenerations and wave functions for the first three energy levels (ground state and first two excited states) of a system of two identical particles with spin , which move in a one- dimensional infinite well of size . Find corrections of energies to first order in if an attracting potential of contact is added. Show that in the case of "spinless" fermions, the previous perturbation has no effect. Step by step process with good handwriting,...
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...
Quantum Mechanics. Find the energies, degenerations and wave functions for the first three energy levels (ground...
Quantum Mechanics.
Find the energies, degenerations and wave functions for the first
three energy levels (ground state
and first two excited states) of a system of two identical
particles with spin , which move in a
one-
dimensional infinite well of size .
Find corrections of energies to first order in if an
attracting potential of contact
is added.
Show that in the case of "spinless" fermions, the previous
perturbation has no effect.
Step by step process with good handwriting,...
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...
[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:(")...
5. (a) State the strong form of the Pauli Principle which describes the symmetry properties of...
5. (a) State the strong form of the Pauli Principle which describes the symmetry properties of a two-particle state function under exchange of the particles, and explain the difference between fermions and bosons. 2 marks) (b) By constructing appropriate two-particle state functions show why it is that no two identical fermions can occupy the same spatial state, i.e. a state specified by quantum numbers n, 1, mi, unless they have different spins. 4 marks) (c) Use the Bohr-Stoner principle (or...
Q1) Consider 2.dimensional infinite "well" with the potential otherwise The stationary states are ny = (a)...
Q1) Consider 2.dimensional infinite "well" with the potential otherwise The stationary states are ny = (a) sin ( x) sin (y,) The corresponding energies are n) , 123 Note that the ground state, ?11 is nondegenerate with the energy E00)-E1)-' r' Now introduce the perturbation, given by the shaded region in the figure ma AH,-{Vo, if 0<x otherwise y<a/2 (a) What is the energy of the 1.st excited state of the unperturbed system? What is its degree of degeneracy,v? (b)...
1-r' Problem 16.12 (30 pts) This chapter examines the two-state system but consider instead the infinite-state...
1-r' Problem 16.12 (30 pts) This chapter examines the two-state system but consider instead the infinite-state system consisting of N non-interacting particles. Each particle i can be in one of an infinite number of states designated by an integer, n; = 0,1,2, .... The energy of particle i is given by a = en; where e is a constant. Note: you may need the series sum Li-ori = a) If the particles are distinguishable, compute QIT,N) and A(T,N) for this...
2. Two noninteracting particles, each of mass m, are in the 1-D harmonic oscillator potential Describe their ground state and the next two excited states, that is, their corresponding (i) wave functions, (i) energy eigenvalues, and (ii) degeneracies if two particles are (a) distinguishable particles, (b) the identical bosons, and (c) the identical fermions. 3. Suppose one particle is in the ground state, and the other is in the first excited state for tw particles described in Prob. 2. Calculate...
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.
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...
Quantum Mechanics.
Find the energies, degenerations and wave functions for the first
three energy levels (ground state
and first two excited states) of a system of two identical
particles with spin , which move in a
one-
dimensional infinite well of size .
Find corrections of energies to first order in if an
attracting potential of contact
is added.
Show that in the case of "spinless" fermions, the previous
perturbation has no effect.
Step by step process with good handwriting,...
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...
[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:(")...
5. (a) State the strong form of the Pauli Principle which describes the symmetry properties of a two-particle state function under exchange of the particles, and explain the difference between fermions and bosons. 2 marks) (b) By constructing appropriate two-particle state functions show why it is that no two identical fermions can occupy the same spatial state, i.e. a state specified by quantum numbers n, 1, mi, unless they have different spins. 4 marks) (c) Use the Bohr-Stoner principle (or...
Q1) Consider 2.dimensional infinite "well" with the potential otherwise The stationary states are ny = (a) sin ( x) sin (y,) The corresponding energies are n) , 123 Note that the ground state, ?11 is nondegenerate with the energy E00)-E1)-' r' Now introduce the perturbation, given by the shaded region in the figure ma AH,-{Vo, if 0<x otherwise y<a/2 (a) What is the energy of the 1.st excited state of the unperturbed system? What is its degree of degeneracy,v? (b)...
1-r' Problem 16.12 (30 pts) This chapter examines the two-state system but consider instead the infinite-state system consisting of N non-interacting particles. Each particle i can be in one of an infinite number of states designated by an integer, n; = 0,1,2, .... The energy of particle i is given by a = en; where e is a constant. Note: you may need the series sum Li-ori = a) If the particles are distinguishable, compute QIT,N) and A(T,N) for this...
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