because for 2 independent(non-interacting) variables A and B,
Also, single particle wavefunctions are
Now, lets begin!
(a) If particles are distinguishable (which is a non-realistic situation for quantum particles, but anyway) there is no condition on the overall wavefunctions, hence we have simply
Note: Because we can label the particle(possible for distinguishable), I have explicitly chosen "1" in n-th state and "2" in l-th state.
Some simplification can be done without actually expanding
(b,c) Can't label them anymore.
+ for Bosons and - for Fermions. On doing the calculation you will get the same answer as (a)
2096) Two noninteracting particles 1 and 2, each of mass m, are in a 1-D infinite...
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...
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...
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
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...
Seven identical particles are placed in a one-dimensional well with infinite potential: the spatial size of the well is L = 1 nm. Calculate the energy of the base state for the system, if the particles are a) electrons b) pions (which are bosons) with mass = 264 and electron mass
4) (2096) For an electron in a one-dimensional infinite square well of width L, find (a) (5%) < x >, (b) (5%) < x2 >, and (c) (5%) Δ). (d) (5%) What is the probability of finding the electron between x = 0.2 L and x = 0.4 L if the electron is in n=5 state
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...
Consider an ideal gas of noninteracting bosons of mass m 0 in 3-D. 1. The fugacity z-eß-c"/hT of the gas can be expanded as a polynomial of the density ρ(-1/v yv): Find Ao, A, and A2. Useful formula: /2(e)+ .. 2. Ί1Kjaessme can bc expanded as The pre where po-is the pressure of a classicla ideal gas Without any calculation, determine the sign of B2, and explain your reason. Calculate B2 Sketch B2 as a function of temperature Consider an...
For these graphs, they are wave functions of identical particles that are within an infinite square well and their width is a. a.)What is the most probable value of the energy for each wave function and which state has the largest probable energy? b.) Which of these states has the largest expectation value of the energy? (this part can be done without calculating the expectation value of the energy) Vi Aax V2 0 elsewhere elsewhere a x Vi Aax V2...