Homework Answers
The given wavefunction is,
replacing x1 by x2, we get
Therefore,
i.e., the wavefunction is antysymmetric.
Hence, the particles are fermions.
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the wavefunction 4. Given two identical particles 1 & 2, and two quantum states ve and...
Problem 1. Consider a system of three identical particles. Each particle has 5 quantum states with...
Problem 1. Consider a system of three identical particles. Each particle has 5 quantum states with energies 0, ε, 2E, 3E, 4E. For distinguishable particles, calculate the number of quantum states where (1) three particles are in the same single-particle state, (2) only two particles are in the same single-particle state, and (3) no two particles are in the same single-particle state. Problem 2. For fermions, (1) calculate the total number of quantum states, and (2) the number of states...
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...
[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. Consider a quantum system comprising three indistinguishable particles which can occupy only three individual-partic...
1. Consider a quantum system comprising three indistinguishable particles which can occupy only three individual-particle energy levels, with energies ε,-0, ε,-2e and ε,-3. The system is in thermal equilibrium at temperature T. Suppose the particles are bosons with integer spin. i) How many states do you expect this system to have? Justify your answer [2 marks] (ii) Make a table showing, for each state of this system, the energy of the state, the number of particles (M, M,, N) with...
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.
Question 5 [2 + 2 + 2 + 3 + 3 = 12 marks ] (a)...
Question 5 [2 + 2 + 2 + 3 + 3 = 12 marks ] (a) Briefly explain why we cannot find simultaneous eigenfunctions of Lg, Ly and Lz. An electron in a hydrogen atom is in the n = 2 state. Ignoring spin, write down the list of possible quantum numbers {n, l, m}. (b) For two qubits briefly explain, giving examples, the difference between a product state and an entangled state (c) Consider a system of identical bosons...
16 , Eo Problem 1 (8 pts): An experimentalist is examining a kind of non-interacting identical...
16 , Eo Problem 1 (8 pts): An experimentalist is examining a kind of non-interacting identical particles that could be either spinless bosons or spin-half fermions by putting a number of them inside a potential and measuring the energy levels of the system, but without being able to resolve the degeneracy of each level. Energy levels do not depend on the particles' spin. The following values of the energy are observed: No particles: 0 -5€ 1 particle: E, 2E, 5E...
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.
[ 2 + 2 + 2 + 3 + 3 = 12 marks ] Question 5 (a) Briefly explain why we cannot find simultaneous eigenfunctions of Lt, L...
[ 2 + 2 + 2 + 3 + 3 = 12 marks ] Question 5 (a) Briefly explain why we cannot find simultaneous eigenfunctions of Lt, L, and Lz. An electron in a hydrogen atom is in the n = 2 state. Ignoring spin, write down the list of possible quantum numbers {n, l, m} (b) For two qubits briefly explain, giving examples, the difference between a product state and an entangled state (c) Consider a system of identical...
Isn't it distinguishable? Because they do not interact with each other. But the answer of boson...
Isn't it distinguishable? Because they do not interact
with each other.
But the answer of boson and fermion are different each
other..
How can I get the answer?
8. Two identical particles of mass m interact with an external potential which is harmonic, but they do not interact with each other. The Hamiltonian of the system is À = pi + P + mw€ (zł + x2). H = 2m + 2m + 2 What are the energy levels of...
Problem 1. Consider a system of three identical particles. Each particle has 5 quantum states with energies 0, ε, 2E, 3E, 4E. For distinguishable particles, calculate the number of quantum states where (1) three particles are in the same single-particle state, (2) only two particles are in the same single-particle state, and (3) no two particles are in the same single-particle state. Problem 2. For fermions, (1) calculate the total number of quantum states, and (2) the number of states...
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...
[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. Consider a quantum system comprising three indistinguishable particles which can occupy only three individual-particle energy levels, with energies ε,-0, ε,-2e and ε,-3. The system is in thermal equilibrium at temperature T. Suppose the particles are bosons with integer spin. i) How many states do you expect this system to have? Justify your answer [2 marks] (ii) Make a table showing, for each state of this system, the energy of the state, the number of particles (M, M,, N) with...
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.
Question 5 [2 + 2 + 2 + 3 + 3 = 12 marks ] (a) Briefly explain why we cannot find simultaneous eigenfunctions of Lg, Ly and Lz. An electron in a hydrogen atom is in the n = 2 state. Ignoring spin, write down the list of possible quantum numbers {n, l, m}. (b) For two qubits briefly explain, giving examples, the difference between a product state and an entangled state (c) Consider a system of identical bosons...
16 , Eo Problem 1 (8 pts): An experimentalist is examining a kind of non-interacting identical particles that could be either spinless bosons or spin-half fermions by putting a number of them inside a potential and measuring the energy levels of the system, but without being able to resolve the degeneracy of each level. Energy levels do not depend on the particles' spin. The following values of the energy are observed: No particles: 0 -5€ 1 particle: E, 2E, 5E...
[ 2 + 2 + 2 + 3 + 3 = 12 marks ] Question 5 (a) Briefly explain why we cannot find simultaneous eigenfunctions of Lt, L, and Lz. An electron in a hydrogen atom is in the n = 2 state. Ignoring spin, write down the list of possible quantum numbers {n, l, m} (b) For two qubits briefly explain, giving examples, the difference between a product state and an entangled state (c) Consider a system of identical...
Isn't it distinguishable? Because they do not interact
with each other.
But the answer of boson and fermion are different each
other..
How can I get the answer?
8. Two identical particles of mass m interact with an external potential which is harmonic, but they do not interact with each other. The Hamiltonian of the system is À = pi + P + mw€ (zł + x2). H = 2m + 2m + 2 What are the energy levels of...
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