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2. Bosons in a s.h.o. Consider a gas of bosons in a 3D harmonic potential, with...
Statistical_Mechanics 2 20 points) 2D ideal Fermi gas 24 Consider an ideal Fermi gas in 2D....
Statistical_Mechanics
2 20 points) 2D ideal Fermi gas 24 Consider an ideal Fermi gas in 2D. It is contained in an area of dimensions L x L. The particle mass is m. (a) Find the density of states D(e) N/L2 (b) Find the Fermi energy as a function of the particle density n = (c) Find the total energy as a function of the Fermi energy ef. (d) Find the chemical potential u as a function of n and T....
Estimate the ground-state energy of a one-dimensional simple harmonic oscillator using (50) = e-a-l as a...
Estimate the ground-state energy of a one-dimensional simple harmonic oscillator using (50) = e-a-l as a trial function with a to be varied. For a simple harmonic oscillator we have H + jmwºr? Recall that, for the variational method, the trial function (HO) gives an expectation value of H such that (016) > Eo, where Eo is the ground state energy. You may use: n! dH() ||= TH(c) – z[1 – H(r)], 8(2), dx S." arcade an+1 where (x) and...
1. Consider a charged particle bound in the harmonic oscillator potential V(x) = mw x2. A...
1. Consider a charged particle bound in the harmonic oscillator potential V(x) = mw x2. A weak electric field is applied to the system such that the potential energy, U(X), now has an extra term: V(x) = -qEx. We write the full Hamiltonian as H = Ho +V(x) where Ho = Px +mw x2 V(x) = –qEx. (a) Write down the unperturbed energies, EO. (b) Find the first-order correction to E . (c) Calculate the second-order correction to E ....
1. Consider a one-dimensional simple harmonic oscillator. We know that the total energy (E) has values:...
1. Consider a one-dimensional simple harmonic oscillator. We know that the total energy (E) has values: Here the angular frequency (o) corresponds to the freshman physics value of [spring constant/massja and (n) can be 0, 1,2, any non-negative integer. We know that the total energy is a measurable, observable quantity. The total energy includes the kinetic energy and the potential energy. Please explain whether or not the kinetic energy and the potential energy can both be measured at the same...
Question 3 Consider the one-dimensional harmonic oscillator, and denote its properly normalised e...
ONLY (e) (f) NEEDED THANK YOU :)
Question 3 Consider the one-dimensional harmonic oscillator, and denote its properly normalised energy eigenstates by { | n〉, n = 0, 1, 2, 3, . . .). Define the state where α is a complex number, and C is a normalisation constant. (a) Use a Campbell-Baker-Hausdorff relation (or otherwise) to show that In other words, | α > is an eigenstate of the (non-Hermitian) lowering operator with (complex) eigenvalue α. (b) During lectures...
BOX 5.1 The Polar Coordinate Basis Consider ordinary polar coordinates r and 0 (see figure 5.3)....
BOX 5.1 The Polar Coordinate Basis Consider ordinary polar coordinates r and 0 (see figure 5.3). Note that the distance between two points with the same r coordinate but separated by an infinitesimal step do in 0 is r do (by the definition of angle). So there are (at least) two ways to define a basis vector for the direction (which we define to be tangent to the r = constant curve): (1) we could define a basis vector es...
The interaction potential between 2 particles that generates the desired effect of particles coming together, sticking,...
The interaction potential between 2 particles that generates the desired effect of particles coming together, sticking, and bouncing off if knocked with sufficient energy is given by the Lennard-Jones (LJ) interaction potential. This potential as a function of distance r between any 2 particles is: U(n) = 4e (9)"-09)) (1) Consider a solution of gold nanoparticles of diameter o = 2 nm. Take e 1 eV. Note that 1 eV = 1.6 x 10-19 Joules. 2. At what distance between...
1. Anharmonic oscillator. Hydrogen bromide, 'HiBr, vibrates approximately according to a Morse potential VM(r) = Dell-e-w2De)1/2 (r-rej2 with De= 4.8 10 eV, re= 1.4 1 44Ă, and k= 408.4 N m-1. Wit...
1. Anharmonic oscillator. Hydrogen bromide, 'HiBr, vibrates approximately according to a Morse potential VM(r) = Dell-e-w2De)1/2 (r-rej2 with De= 4.8 10 eV, re= 1.4 1 44Ă, and k= 408.4 N m-1. With ω,-VRA, the energies of the stationary states in a Morse potential are En (hwo) 4D ho(n+ 1/2)- (n + 1/2)2. (A) On the same graph, plot the Morse potential and the harmonic potential as a function of bond length (from 0.7 Te to 2 re).(B) Describe the differences....
4) Consider n data points with 2 covariates and observation {xi,i, Vi,2, yi); i -1,... ,n,...
4) Consider n data points with 2 covariates and observation {xi,i, Vi,2, yi); i -1,... ,n, where yi 's are indicator variable for the experiment that is if a particular medicine is effective on some individual. Here, xi1 and ri.2 are age and blood pressure of i th individual, respectively. Our assumption is that the log odds ratio follows a linear model. That is p-P(i-1) and 10i b) What should be a good estimator for ?,A, e) Suppose. On, A,n...
Problem 5. (30 points) Consider a Harmonic oscillator with H that H=(ata + 1 / 2)ho,...
Problem 5. (30 points) Consider a Harmonic oscillator with H that H=(ata + 1 / 2)ho, where a=dma)X + i (a) (4 points) Show P, and a x 2h 2h 2moh P. Show also 2moh that [a, a]-l. (b) (6 points) Starting from the commuters la, HJand la', A), where H-H(h) show that the eigenvalues of Hare e,=(n+1/2) for n-0, 1,2, Show also that alm)-nln-l), and a l). (( points) Find the normalized ground state wavefunction by projecting alo)-0 on...
Statistical_Mechanics
2 20 points) 2D ideal Fermi gas 24 Consider an ideal Fermi gas in 2D. It is contained in an area of dimensions L x L. The particle mass is m. (a) Find the density of states D(e) N/L2 (b) Find the Fermi energy as a function of the particle density n = (c) Find the total energy as a function of the Fermi energy ef. (d) Find the chemical potential u as a function of n and T....
Estimate the ground-state energy of a one-dimensional simple harmonic oscillator using (50) = e-a-l as a trial function with a to be varied. For a simple harmonic oscillator we have H + jmwºr? Recall that, for the variational method, the trial function (HO) gives an expectation value of H such that (016) > Eo, where Eo is the ground state energy. You may use: n! dH() ||= TH(c) – z[1 – H(r)], 8(2), dx S." arcade an+1 where (x) and...
1. Consider a charged particle bound in the harmonic oscillator potential V(x) = mw x2. A weak electric field is applied to the system such that the potential energy, U(X), now has an extra term: V(x) = -qEx. We write the full Hamiltonian as H = Ho +V(x) where Ho = Px +mw x2 V(x) = –qEx. (a) Write down the unperturbed energies, EO. (b) Find the first-order correction to E . (c) Calculate the second-order correction to E ....
1. Consider a one-dimensional simple harmonic oscillator. We know that the total energy (E) has values: Here the angular frequency (o) corresponds to the freshman physics value of [spring constant/massja and (n) can be 0, 1,2, any non-negative integer. We know that the total energy is a measurable, observable quantity. The total energy includes the kinetic energy and the potential energy. Please explain whether or not the kinetic energy and the potential energy can both be measured at the same...
ONLY (e) (f) NEEDED THANK YOU :)
Question 3 Consider the one-dimensional harmonic oscillator, and denote its properly normalised energy eigenstates by { | n〉, n = 0, 1, 2, 3, . . .). Define the state where α is a complex number, and C is a normalisation constant. (a) Use a Campbell-Baker-Hausdorff relation (or otherwise) to show that In other words, | α > is an eigenstate of the (non-Hermitian) lowering operator with (complex) eigenvalue α. (b) During lectures...
BOX 5.1 The Polar Coordinate Basis Consider ordinary polar coordinates r and 0 (see figure 5.3). Note that the distance between two points with the same r coordinate but separated by an infinitesimal step do in 0 is r do (by the definition of angle). So there are (at least) two ways to define a basis vector for the direction (which we define to be tangent to the r = constant curve): (1) we could define a basis vector es...
The interaction potential between 2 particles that generates the desired effect of particles coming together, sticking, and bouncing off if knocked with sufficient energy is given by the Lennard-Jones (LJ) interaction potential. This potential as a function of distance r between any 2 particles is: U(n) = 4e (9)"-09)) (1) Consider a solution of gold nanoparticles of diameter o = 2 nm. Take e 1 eV. Note that 1 eV = 1.6 x 10-19 Joules. 2. At what distance between...
1. Anharmonic oscillator. Hydrogen bromide, 'HiBr, vibrates approximately according to a Morse potential VM(r) = Dell-e-w2De)1/2 (r-rej2 with De= 4.8 10 eV, re= 1.4 1 44Ă, and k= 408.4 N m-1. With ω,-VRA, the energies of the stationary states in a Morse potential are En (hwo) 4D ho(n+ 1/2)- (n + 1/2)2. (A) On the same graph, plot the Morse potential and the harmonic potential as a function of bond length (from 0.7 Te to 2 re).(B) Describe the differences....
4) Consider n data points with 2 covariates and observation {xi,i, Vi,2, yi); i -1,... ,n, where yi 's are indicator variable for the experiment that is if a particular medicine is effective on some individual. Here, xi1 and ri.2 are age and blood pressure of i th individual, respectively. Our assumption is that the log odds ratio follows a linear model. That is p-P(i-1) and 10i b) What should be a good estimator for ?,A, e) Suppose. On, A,n...
Problem 5. (30 points) Consider a Harmonic oscillator with H that H=(ata + 1 / 2)ho, where a=dma)X + i (a) (4 points) Show P, and a x 2h 2h 2moh P. Show also 2moh that [a, a]-l. (b) (6 points) Starting from the commuters la, HJand la', A), where H-H(h) show that the eigenvalues of Hare e,=(n+1/2) for n-0, 1,2, Show also that alm)-nln-l), and a l). (( points) Find the normalized ground state wavefunction by projecting alo)-0 on...
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