Give the time dependence of the state function for (a) a system which is in an...
Give the time dependence of the state function for
(a) a system which is in an eigenstate of its Hamiltonian with eigenvalue B
(b) a system which is an unspecified mixture of an n-fold degenerate set of eigenstates of its Hamiltonian with eigenvalue B
Homework Answers
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Give the time dependence of the state function for
(a) a system which is in an...
Hello, I need help with a problem for my Quantum Mechanics class. Please explain as if...
Hello, I need help with a problem for my Quantum Mechanics class. Please explain as if I am learning for the first time. I want to be able to understand and do problems like this on my own. Thank you in advance for your help! The infinite square well has solutions that are very familiar to us from previous physics classes. However, in this class we learn that a quantum state of the system can be in a superposition state...
Exercise 8: Time dependence of a two-level system Consider a two-level system with stationary states a and b with unper...
Exercise 8: Time dependence of a two-level system Consider a two-level system with stationary states a and b with unperturbed energies and E() and corresponding eigenfunctions ф ) and 4°, respectively. Assume E ) > E 0. such that the Bohr angular frequency wi(EEis positive. A time-independent perturbation V is switched on at time t 0 a) Write down the coupled set of equations for the coefficients ca(t) and c(t) of the wave function of the system: Note that we...
Section IV 13. Explain what happens to the molecular partition function () in the limits of Tinfinity and T→0 for a system that has a five-fold degenerate ground state, a nondegenerate first exci...
Section IV 13. Explain what happens to the molecular partition function () in the limits of Tinfinity and T→0 for a system that has a five-fold degenerate ground state, a nondegenerate first excited state, and a three-fold degenerate second excited state. There are no other states in this system. (5 pts)
Section IV 13. Explain what happens to the molecular partition function () in the limits of Tinfinity and T→0 for a system that has a five-fold degenerate ground state,...
2. The hydrogen atom [8 marks] The time-independent Schrödinger equation for the hydrogen atom in...
2. The hydrogen atom [8 marks] The time-independent Schrödinger equation for the hydrogen atom in the spherical coordinate representation is where ao-top- 0.5298 10-10rn is the Bohr radius, and μ is the electon-proton reduced mass. Here, the square of the angular momentum operator L2 in the spherical coordinate representation is given by: 2 (2.2) sin θー sin θ 00 The form of the Schrödinger equation means that all energy eigenstates separate into radial and angular motion, and we can write...
3. Ehrenfest's Theorem states that dA i for an observable A and a time-independent Hamiltonian H....
3. Ehrenfest's Theorem states that dA i for an observable A and a time-independent Hamiltonian H. Consider a QM system in 1D with time-independent Hamiltonian H 2 V(x). Use Ehrenfest's Theorem to determine Jlx) and p). What is 듦(z)? 4. A projection operator Pn is defined by where there is no summation Ση implied, and the states n> are a complete set of oth onormal states (a) Show that Pa satisfies 2 (b) Show that Pn acting on an arbitrary...
qm 2019.3 3. The Hamiltonian corresponding to the magnetic interaction of a spin 1/2 particle with...
qm 2019.3
3. The Hamiltonian corresponding to the magnetic interaction of a spin 1/2 particle with charge e and mass m in a magnetic field B is À eB B. Ŝ, m where Ŝ are the spin angular momentum operators. You should make use of expres- sions for the spin operators that are given at the end of the question. (i) Write down the energy eigenvalue equation for this particle in a field directed along the y axis, i.e. B...
Please answer the question in full and show all work. We have seen that the absolute...
Please answer the question in full and show all work.
We have seen that the absolute square of the wave function VI,t) can be interpreted as the probability density for the location of the particle at time t. We have also seen that a particle's quantum state can be represented as a linear combination of eigenstates of a physical observable Q: V) SIT) where Q n ) = qn|n) and represents the probability to find the particle in the eigenstate...
1. (30 points). Coupled spins. Spin-1/2 particles A and B evolve under the influence of the follo...
1. (30 points). Coupled spins. Spin-1/2 particles A and B evolve under the influence of the following Hamiltonian (for simplicity takeh-1 so that energies are expressed in frequency units): We work in the uncoupled basis aba) Ib), where a,b E 0,1 and where states 0) (1)) correspond to single spins aligned (antialigned) with the z-direction. As we discussed in lecture, the eigenstates of the Hamiltonian are 100), 111), and 2-1/2 (101) 110)). a. We prepare the initial state t01). Since...
PLEASE COMPLETE B) and stay tuned for my following 2 questions where I will ask part c) and d). P...
PLEASE COMPLETE B) and stay tuned for my
following 2 questions where I will ask part c) and d). Part a) has
already been posted.
The lowest energy state of a hydrogen-like atom has total angular momentum J-1/2 (from the l-O orbital angular momentum and the electron spin s 1/2). Furthermore, the nucleus also has a spin, conventionally labeled I (for hydrogen, this is the proton spin, 1 1/2). This spin leads to an additional degeneracy. For example, in the...
6. a) Calculate the expectation value of x as a function of time for an electron...
6. a) Calculate the expectation value of x as a function of time for an electron in a state that is a (normalized) equal mixture of the ground state and 1st excited state of a 1D HO b) Graph x vs time for the case k = 1 eV/nm2. What is its value at t=0? What is the period of the oscillation in femtoseconds? For the one-dimensional (1D) harmonic oscillator (HO) the potential energy function has the form V(a) k2/2,...
Hello, I need help with a problem for my Quantum Mechanics class. Please explain as if I am learning for the first time. I want to be able to understand and do problems like this on my own. Thank you in advance for your help! The infinite square well has solutions that are very familiar to us from previous physics classes. However, in this class we learn that a quantum state of the system can be in a superposition state...
Exercise 8: Time dependence of a two-level system Consider a two-level system with stationary states a and b with unperturbed energies and E() and corresponding eigenfunctions ф ) and 4°, respectively. Assume E ) > E 0. such that the Bohr angular frequency wi(EEis positive. A time-independent perturbation V is switched on at time t 0 a) Write down the coupled set of equations for the coefficients ca(t) and c(t) of the wave function of the system: Note that we...
Section IV 13. Explain what happens to the molecular partition function () in the limits of Tinfinity and T→0 for a system that has a five-fold degenerate ground state, a nondegenerate first excited state, and a three-fold degenerate second excited state. There are no other states in this system. (5 pts)
Section IV 13. Explain what happens to the molecular partition function () in the limits of Tinfinity and T→0 for a system that has a five-fold degenerate ground state,...
2. The hydrogen atom [8 marks] The time-independent Schrödinger equation for the hydrogen atom in the spherical coordinate representation is where ao-top- 0.5298 10-10rn is the Bohr radius, and μ is the electon-proton reduced mass. Here, the square of the angular momentum operator L2 in the spherical coordinate representation is given by: 2 (2.2) sin θー sin θ 00 The form of the Schrödinger equation means that all energy eigenstates separate into radial and angular motion, and we can write...
3. Ehrenfest's Theorem states that dA i for an observable A and a time-independent Hamiltonian H. Consider a QM system in 1D with time-independent Hamiltonian H 2 V(x). Use Ehrenfest's Theorem to determine Jlx) and p). What is 듦(z)? 4. A projection operator Pn is defined by where there is no summation Ση implied, and the states n> are a complete set of oth onormal states (a) Show that Pa satisfies 2 (b) Show that Pn acting on an arbitrary...
qm 2019.3
3. The Hamiltonian corresponding to the magnetic interaction of a spin 1/2 particle with charge e and mass m in a magnetic field B is À eB B. Ŝ, m where Ŝ are the spin angular momentum operators. You should make use of expres- sions for the spin operators that are given at the end of the question. (i) Write down the energy eigenvalue equation for this particle in a field directed along the y axis, i.e. B...
Please answer the question in full and show all work.
We have seen that the absolute square of the wave function VI,t) can be interpreted as the probability density for the location of the particle at time t. We have also seen that a particle's quantum state can be represented as a linear combination of eigenstates of a physical observable Q: V) SIT) where Q n ) = qn|n) and represents the probability to find the particle in the eigenstate...
1. (30 points). Coupled spins. Spin-1/2 particles A and B evolve under the influence of the following Hamiltonian (for simplicity takeh-1 so that energies are expressed in frequency units): We work in the uncoupled basis aba) Ib), where a,b E 0,1 and where states 0) (1)) correspond to single spins aligned (antialigned) with the z-direction. As we discussed in lecture, the eigenstates of the Hamiltonian are 100), 111), and 2-1/2 (101) 110)). a. We prepare the initial state t01). Since...
PLEASE COMPLETE B) and stay tuned for my
following 2 questions where I will ask part c) and d). Part a) has
already been posted.
The lowest energy state of a hydrogen-like atom has total angular momentum J-1/2 (from the l-O orbital angular momentum and the electron spin s 1/2). Furthermore, the nucleus also has a spin, conventionally labeled I (for hydrogen, this is the proton spin, 1 1/2). This spin leads to an additional degeneracy. For example, in the...
6. a) Calculate the expectation value of x as a function of time for an electron in a state that is a (normalized) equal mixture of the ground state and 1st excited state of a 1D HO b) Graph x vs time for the case k = 1 eV/nm2. What is its value at t=0? What is the period of the oscillation in femtoseconds? For the one-dimensional (1D) harmonic oscillator (HO) the potential energy function has the form V(a) k2/2,...
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