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3- A one-dimensional harmonic oscillator wave function is ψ(x) = Axe-bx2 a) Find the total energy...
4. (20 points). Consider a quantum harmonic oscillator with characteristic frequency w. The syste...
4. (20 points). Consider a quantum harmonic oscillator with characteristic frequency w. The system is in thermal equilibrium at temperature T. The oscillator is described by the following density matrix: A exp kaT where H is the usual harmonic oscillator Hamiltonian and kB is Boltzmann's constant. Working in the Fock (photon number) basis: a. Find the diagonal elements of ρ b. Determine the normalization constant A. c. Calculate the expectation value of energy (E
4. (20 points). Consider a quantum...
4. (20 points) Harmonic Oscillator The ground state wave function of a simple harmonic oscillator is...
4. (20 points) Harmonic Oscillator The ground state wave function of a simple harmonic oscillator is (a) = Ae-42", where a = (a) Using the normalization condition, obtain the constant A. (b) Find (c), (), and Az, using the result of A obtained in (a). Again, A.= V(32) - (2) (c) Find (p) and Ap. For the latter, you need to evaluate (p). Hint: For a harmonic oscillator, the time-averaged kinetic energy is equal to the time-averaged potential energy, and...
Question A2: Coherent states of the harmonic oscillator Consider a one-dimensional harmonic oscillator with the Hamiltonian...
Question A2: Coherent states of the harmonic oscillator Consider a one-dimensional harmonic oscillator with the Hamiltonian 12 12 m2 H = -2m d. 2+ 2 Here m and w are the mass and frequency, respectively. Consider a time-dependent wave function of the form <(x,t) = C'exp (-a(x – 9(t)+ ik(t)z +io(t)), where a and C are positive constants, and g(t), k(t), and o(t) are real functions of time t. 1. Express C in terms of a. [2 marks] 2. By...
Q3) A particle in the harmonic oscillator potential has the initial normalized wave function Ψ(?, 0)...
Q3) A particle in the harmonic oscillator potential has the initial normalized wave function Ψ(?, 0) = 1 /√5 [2 ?₁ (?) + ?₂ (?)] where ?1 and ?2 are the eigenfunctions of the oscillator Hamiltonian for ? = 1,2 states. a) Write down the expression for Ψ(?,?). b) Calculate the probability density ℙ(?,?) = |Ψ(?,?)| ² . Express it as a sinusoidal function of time. To simplify the result, let ? ≡ (?² ℏ)/ 2??² . c) Calculate 〈?〉...
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...
7 Harmonic oscillator in "energy space" Consider the harmonic oscillator in "energy space", i.e., in terms...
7 Harmonic oscillator in "energy space" Consider the harmonic oscillator in "energy space", i.e., in terms of the basis of eigenvectors n) of the harmonic oscillator Hamiltonian, with Hn) -hwn1/2)]n). We computed these in terms of wavefunctions in position space, ie. pn(x)-(zln), but we can also work purely in terms of the abstract energy eigenvectors in Dirac notation. PS9.pdf 1. You computed the matrix elements 〈nleln) on an earlier problem set. Now find (nn) for general n,n' 2. Find the...
A particle with mass m is in a one-dimensional simple harmonic oscillator potential. At time t = 0 it is described by the state where lo and l) are normalised energy eigenfunctions corresponding to e...
A particle with mass m is in a one-dimensional simple harmonic oscillator potential. At time t = 0 it is described by the state where lo and l) are normalised energy eigenfunctions corresponding to energies E and Ey and b and c are real constants. (a) Find b and c so that (x) is as large as possible. b) Write down the wavefunction of this particle at a time t later c)Caleulate (x) for the particle at time t (d)...
2.3 The wave function of a harmonic oscillator with the parameters m and w is a...
2.3 The wave function of a harmonic oscillator with the parameters m and w is a superposition of n 0 and n 2 stationary states o(Z,x ), 〈p), 〈d V(x)〉 (5 points) (a) compute the expectation values (r (b) find the expectation value and the variance of the total energy, Which value of the energy you can actually get when doing measurements and with which probability? (5 points)
2. Prove Find the value of the normalization constant A for the wave function y Axe 2. Prove Find the value of the normalization constant A for the wave function y Axe
2. Prove Find the value of the normalization constant A for the wave function y Axe
2. Prove Find the value of the normalization constant A for the wave function y Axe
Consider the harmonic oscillator wave function 1/4 where α = (-)"*. Here k, is the stiffness...
Consider the harmonic oscillator wave function 1/4 where α = (-)"*. Here k, is the stiffness coefficient of the oscillator and m is mass. Recall that the oscillation frequency iso,s:,k, / m In class we showed that Ψ0(x) Is an eigenfunction of the Hamiltonian, with an eigenvalue Eo (1/2)ha a) Normalize the wave function in Eq.(1) b) Graph the probability density. Note that a has units of length and measures the "width" of the wave function. It's easier to use...
4. (20 points). Consider a quantum harmonic oscillator with characteristic frequency w. The system is in thermal equilibrium at temperature T. The oscillator is described by the following density matrix: A exp kaT where H is the usual harmonic oscillator Hamiltonian and kB is Boltzmann's constant. Working in the Fock (photon number) basis: a. Find the diagonal elements of ρ b. Determine the normalization constant A. c. Calculate the expectation value of energy (E
4. (20 points). Consider a quantum...
4. (20 points) Harmonic Oscillator The ground state wave function of a simple harmonic oscillator is (a) = Ae-42", where a = (a) Using the normalization condition, obtain the constant A. (b) Find (c), (), and Az, using the result of A obtained in (a). Again, A.= V(32) - (2) (c) Find (p) and Ap. For the latter, you need to evaluate (p). Hint: For a harmonic oscillator, the time-averaged kinetic energy is equal to the time-averaged potential energy, and...
Question A2: Coherent states of the harmonic oscillator Consider a one-dimensional harmonic oscillator with the Hamiltonian 12 12 m2 H = -2m d. 2+ 2 Here m and w are the mass and frequency, respectively. Consider a time-dependent wave function of the form <(x,t) = C'exp (-a(x – 9(t)+ ik(t)z +io(t)), where a and C are positive constants, and g(t), k(t), and o(t) are real functions of time t. 1. Express C in terms of a. [2 marks] 2. By...
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...
7 Harmonic oscillator in "energy space" Consider the harmonic oscillator in "energy space", i.e., in terms of the basis of eigenvectors n) of the harmonic oscillator Hamiltonian, with Hn) -hwn1/2)]n). We computed these in terms of wavefunctions in position space, ie. pn(x)-(zln), but we can also work purely in terms of the abstract energy eigenvectors in Dirac notation. PS9.pdf 1. You computed the matrix elements 〈nleln) on an earlier problem set. Now find (nn) for general n,n' 2. Find the...
A particle with mass m is in a one-dimensional simple harmonic oscillator potential. At time t = 0 it is described by the state where lo and l) are normalised energy eigenfunctions corresponding to energies E and Ey and b and c are real constants. (a) Find b and c so that (x) is as large as possible. b) Write down the wavefunction of this particle at a time t later c)Caleulate (x) for the particle at time t (d)...
2.3 The wave function of a harmonic oscillator with the parameters m and w is a superposition of n 0 and n 2 stationary states o(Z,x ), 〈p), 〈d V(x)〉 (5 points) (a) compute the expectation values (r (b) find the expectation value and the variance of the total energy, Which value of the energy you can actually get when doing measurements and with which probability? (5 points)
2. Prove Find the value of the normalization constant A for the wave function y Axe
2. Prove Find the value of the normalization constant A for the wave function y Axe
Consider the harmonic oscillator wave function 1/4 where α = (-)"*. Here k, is the stiffness coefficient of the oscillator and m is mass. Recall that the oscillation frequency iso,s:,k, / m In class we showed that Ψ0(x) Is an eigenfunction of the Hamiltonian, with an eigenvalue Eo (1/2)ha a) Normalize the wave function in Eq.(1) b) Graph the probability density. Note that a has units of length and measures the "width" of the wave function. It's easier to use...
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