(f) Show that a wavefunction (x) H()eimhhas (x) has (x) 0 and and (p)muzo. At what time would the classical oscillator have values that match these expectation values?
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The lowest energy wavefunction of the quantum harmonic oscillator has the form (c) Determine σ an...
In the lecture notes, we only solved the TISE for the quantum harmonic oscillator 1 Now,...
In the lecture notes, we only solved the TISE for the quantum harmonic oscillator 1 Now, write down the actual solution of the wavefunction of the quantum harmonic oscillator, i.e. the solution that solves TDSE not TISE. 2. We consider the Quantum Harmonic Oscillator In Heisenberg Picture: (a) Hamiltonian to use is the quantum harmonic oscillator Hamiltonian Solve the Heisenberg equations of motion for the operators X (t) and P(t) where the Calculate the commutator [X(t), X (0)] and show...
3. The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an...
3. The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Consider an electron trapped by a one-dimensional harmonic potential V(x)=-5 mo?x” (where m is the electron mass, o is a constant angular frequency). In this case, the Schrödinger equation takes the following form, **...
1. Quantum harmonic oscillator (a) Derive formula for standard deviation of position measurement ...
1. Quantum harmonic oscillator (a) Derive formula for standard deviation of position measurement on a particle prepared in the ground state of harmonic oscillator. The formula will depend on h, m andw (b) Estimate order of magnitude of the standard deviation in (a) for the LIGO mirror of mass 10 kg and w 1 Hz. (c) A coherent state lo) is defined to be the eigenstate of the lowering operator with eigenvalue a, i.e. à lo)a) Write la) as where...
A harmonic oscillator is in a state such that a measurement of the energy would yield...
A harmonic oscillator is in a state such that a measurement of the energy would yield either Eo=(1/2ho 1/3 of the time or E1-(3/2)hw, 2/3 of the time time-dependent wave function which describes this state. (b) Find the average position of the particle (x(t)). (c) Find the expectation value of the energy. Given that E(H), is the energy associate with this state constant or it oscillates in time between Eo and E1?
Quantum Chemistry. Thx in Advance! 1. For a harmonic oscillator with unit mass and unit frequency,...
Quantum Chemistry. Thx in Advance!
1. For a harmonic oscillator with unit mass and unit frequency, the Schrödinger equation for its eigenfunction is given by where n 0, 1, 2, . . .. Answer the following questions. Given a trial wave function, ?(x)-?000CnUn(x), where expression for the expectation value is is assumed to be real, the Prove that Eo2 h/2 2. Assume that the trial wave function for the ground state eigenfunction in Eq. (1) is ?(x) = cos Xx,...
It can be shown that for a linear harmonic oscillator the expectation value of the potential energy is equal to the expectation value of the kinetic energy, and the expectation values for r and p are...
It can be shown that for a linear harmonic oscillator the
expectation value of the potential energy is equal to the
expectation value of the kinetic energy, and the expectation values
for r and p are clearly both zeros (0) Show that in the lowest
energy state Ain agreement with the uncertainty principle (b)
Confirm that for the higher states (Ax)(Ap) > h/2 .
Problemi 4. ( 8 pts) It can be shown that for a linear harmonic oscillator the...
Problem #1 The explicit wavefunction for a particle in the n-1 state of the quantum harmonic...
Problem #1 The explicit wavefunction for a particle in the n-1 state of the quantum harmonic oscillator is p1(x)- Axe-bx2 where mo 2h and ?1/4 (Note: In last week's homework there was an "h" where there should have been ?. This has been corrected in this week's assignment.) (a) By applying the lowering operator to ),obtain an explicit form for o(x) (i.e. the n-0 wavefunction) (b) By applying the raising operator to x), obtain an explicit form for p2(x) (i.e....
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...
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...
Question no 6.1, statistical physics by Reif Volume 5 Problems 6.1 Phase space of a classical harmonic oscillator The en...
Question no 6.1,
statistical physics by Reif Volume 5
Problems 6.1 Phase space of a classical harmonic oscillator The energy of a one-dimensional harmonic oscillator, whose position coordinate is x and whose momentum is p, is given by where the first term on the right is its kinetic and the second term its potential energy. Here m denotes the mass of the osellating particle and a the spring constant of the restoring force acting on the particle. Consider an ensemble...
In the lecture notes, we only solved the TISE for the quantum harmonic oscillator 1 Now, write down the actual solution of the wavefunction of the quantum harmonic oscillator, i.e. the solution that solves TDSE not TISE. 2. We consider the Quantum Harmonic Oscillator In Heisenberg Picture: (a) Hamiltonian to use is the quantum harmonic oscillator Hamiltonian Solve the Heisenberg equations of motion for the operators X (t) and P(t) where the Calculate the commutator [X(t), X (0)] and show...
3. The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Consider an electron trapped by a one-dimensional harmonic potential V(x)=-5 mo?x” (where m is the electron mass, o is a constant angular frequency). In this case, the Schrödinger equation takes the following form, **...
1. Quantum harmonic oscillator (a) Derive formula for standard deviation of position measurement on a particle prepared in the ground state of harmonic oscillator. The formula will depend on h, m andw (b) Estimate order of magnitude of the standard deviation in (a) for the LIGO mirror of mass 10 kg and w 1 Hz. (c) A coherent state lo) is defined to be the eigenstate of the lowering operator with eigenvalue a, i.e. à lo)a) Write la) as where...
A harmonic oscillator is in a state such that a measurement of the energy would yield either Eo=(1/2ho 1/3 of the time or E1-(3/2)hw, 2/3 of the time time-dependent wave function which describes this state. (b) Find the average position of the particle (x(t)). (c) Find the expectation value of the energy. Given that E(H), is the energy associate with this state constant or it oscillates in time between Eo and E1?
Quantum Chemistry. Thx in Advance!
1. For a harmonic oscillator with unit mass and unit frequency, the Schrödinger equation for its eigenfunction is given by where n 0, 1, 2, . . .. Answer the following questions. Given a trial wave function, ?(x)-?000CnUn(x), where expression for the expectation value is is assumed to be real, the Prove that Eo2 h/2 2. Assume that the trial wave function for the ground state eigenfunction in Eq. (1) is ?(x) = cos Xx,...
It can be shown that for a linear harmonic oscillator the
expectation value of the potential energy is equal to the
expectation value of the kinetic energy, and the expectation values
for r and p are clearly both zeros (0) Show that in the lowest
energy state Ain agreement with the uncertainty principle (b)
Confirm that for the higher states (Ax)(Ap) > h/2 .
Problemi 4. ( 8 pts) It can be shown that for a linear harmonic oscillator the...
Problem #1 The explicit wavefunction for a particle in the n-1 state of the quantum harmonic oscillator is p1(x)- Axe-bx2 where mo 2h and ?1/4 (Note: In last week's homework there was an "h" where there should have been ?. This has been corrected in this week's assignment.) (a) By applying the lowering operator to ),obtain an explicit form for o(x) (i.e. the n-0 wavefunction) (b) By applying the raising operator to x), obtain an explicit form for p2(x) (i.e....
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...
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...
Question no 6.1,
statistical physics by Reif Volume 5
Problems 6.1 Phase space of a classical harmonic oscillator The energy of a one-dimensional harmonic oscillator, whose position coordinate is x and whose momentum is p, is given by where the first term on the right is its kinetic and the second term its potential energy. Here m denotes the mass of the osellating particle and a the spring constant of the restoring force acting on the particle. Consider an ensemble...
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