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4. Consider a particle exposed to the following potential energy (we are using spherical coordinates r,...
#1 A particle of mass, m, moves in a field whose potential energy in spherical coordinates...
#1 A particle of mass, m, moves in a field whose potential energy in spherical coordinates has a 2 , where r and are the standard variables of spherical coordinates and k is a positive constant. Find Hamiltonian and Hamilton's equations of motion for this particle. form of V --k cose
Quantum Mechanics Problem 1. (25) Consider an infinite potential well with the following shape: 0 a/4...
Quantum Mechanics Problem
1. (25) Consider an infinite potential well with the following shape: 0 a/4 3al4 a h2 where 4 Using the ground state wavefunction of the original infinite potential well as a trial function, 2πχ trial = 1-sin- find the approximation of the ground state energy for this system with the variational method. (Note, this question is simplified by considering the two components of the Hamiltonian, and V, on their own) b) If we had used the 1st...
4.2 The potential energy in a MOFSET device near the metal oxide interface is approximately V(x) ...
4.2 The potential energy in a MOFSET device near the metal oxide interface is approximately V(x) - qEx forx > 0 where q is the electron charge, and E is the electric field strength. Use the variational technique to estimate the ground state energy of an electron in this configuration. (Hints: a) use the un-normalized trial function ф(x)-x exp(-ax2)). b) Find the normalized trial wave-function c) Compute the energy functional (i.e. the expectation value of the Hamiltonian for the state...
2. The unational method is an incredibly simple but surprisingly powerful method for understanding the low- energy beha...
2. The unational method is an incredibly simple but surprisingly powerful method for understanding the low- energy behavior of quantu systems. It is used constantly in marny-body physics and in quantum chemistry. The main idea is thst for any physical Hamiltonian, there is a lowest energy state, i.e. the ground state Ipo). All other states (ignoring degeneracy) have higher energy that this one. Therefore we have Therefore, to get an upper bound on the energy of Eo, it suffices to...
A particle of mass m is bound by the spherically-symmetric three-dimensional harmonic- oscillator potential energy ,...
A particle of mass m is bound by the spherically-symmetric three-dimensional harmonic- oscillator potential energy , and ф are the usual spherical coordinates. (a) In the form given above, why is it clear that the potential energy function V) is (b) For this problem, it will be more convenient to express this spherically-symmetric where r , spherically symmetric? A brief answer is sufficient. potential energy in Cartesian coordinates x, y, and z as physically the same potential energy as the...
Use the variational principle to estimate the ground state energy of a particle in the following...
Use the variational principle to estimate the ground state energy of a particle in the following potential: V=cx for x>0 and V=infinity for x<0 Use Dxe^-ax as your trial function and minimize as necessary with respect to a. Assume the constant c is real and greater than zero
a) Use the variational method to estimate the binding energy of a deuteron. Assume that the...
a) Use the variational method to estimate the binding energy of a deuteron. Assume that the potential between the proton and neutron is V(r) = Ae-r/ro where A and ro are constants and use as a trial function W(r) = Ce-Br (4) where C is the normalization constant. b) Consider the Hamiltonian of a nonharmonic oscillator d2 (5) H + x2 + x4 dx2 Use the WKB approximation to find the ground state of the system as x .
Consider a trial wavefunction for an electron in H atom in the following form •(r) =...
Consider a trial wavefunction for an electron in H atom in the following form •(r) = re-ar where a is an adjustable parameter. Optimize a so that you obtain the minimum energy (i.e., find the extremum by imposing (E) = 0). How does the minimum energy compare to the ground state energy of an electron? Hint: n! for a>0 A nEN ne-andc= +1 Integration of function f(r, 0,6) in spherical coordinates: $*$*$* $(1,0, 6)r? sin ødødødr f(,0,)ra sin døddr
2. Potential Inside a Sphere We are interested in the electric potential inside a spherical shell...
2. Potential Inside a Sphere We are interested in the electric potential inside a spherical shell that is radius a and centered on the origin. There are no charges inside the she, so the potential satisfies the Laplace equation, However, there is an external voltage applied to the surface of the shell which holds the potential on the surface to a value which depends on θ: As a result, the potential Ф(r,0) -by symmetry, it does not depend on ф-is...
(a) Use the variational method to estimate the ground state energy of a particle of |mass...
(a) Use the variational method to estimate the ground state energy of a particle of |mass m in a potential Vx)kx, k > 0. (b) Calculate the energy shift in the ground state and in the degenerate 1t excited state of a 2-dimensional harmonic oscillator H(P2P,2/2m m(x +y)due to the perturbation V 2Axy. (20 pts)
(a) Use the variational method to estimate the ground state energy of a particle of |mass m in a potential Vx)kx, k > 0. (b)...
#1 A particle of mass, m, moves in a field whose potential energy in spherical coordinates has a 2 , where r and are the standard variables of spherical coordinates and k is a positive constant. Find Hamiltonian and Hamilton's equations of motion for this particle. form of V --k cose
Quantum Mechanics Problem
1. (25) Consider an infinite potential well with the following shape: 0 a/4 3al4 a h2 where 4 Using the ground state wavefunction of the original infinite potential well as a trial function, 2πχ trial = 1-sin- find the approximation of the ground state energy for this system with the variational method. (Note, this question is simplified by considering the two components of the Hamiltonian, and V, on their own) b) If we had used the 1st...
4.2 The potential energy in a MOFSET device near the metal oxide interface is approximately V(x) - qEx forx > 0 where q is the electron charge, and E is the electric field strength. Use the variational technique to estimate the ground state energy of an electron in this configuration. (Hints: a) use the un-normalized trial function ф(x)-x exp(-ax2)). b) Find the normalized trial wave-function c) Compute the energy functional (i.e. the expectation value of the Hamiltonian for the state...
2. The unational method is an incredibly simple but surprisingly powerful method for understanding the low- energy behavior of quantu systems. It is used constantly in marny-body physics and in quantum chemistry. The main idea is thst for any physical Hamiltonian, there is a lowest energy state, i.e. the ground state Ipo). All other states (ignoring degeneracy) have higher energy that this one. Therefore we have Therefore, to get an upper bound on the energy of Eo, it suffices to...
A particle of mass m is bound by the spherically-symmetric three-dimensional harmonic- oscillator potential energy , and ф are the usual spherical coordinates. (a) In the form given above, why is it clear that the potential energy function V) is (b) For this problem, it will be more convenient to express this spherically-symmetric where r , spherically symmetric? A brief answer is sufficient. potential energy in Cartesian coordinates x, y, and z as physically the same potential energy as the...
a) Use the variational method to estimate the binding energy of a deuteron. Assume that the potential between the proton and neutron is V(r) = Ae-r/ro where A and ro are constants and use as a trial function W(r) = Ce-Br (4) where C is the normalization constant. b) Consider the Hamiltonian of a nonharmonic oscillator d2 (5) H + x2 + x4 dx2 Use the WKB approximation to find the ground state of the system as x .
Consider a trial wavefunction for an electron in H atom in the following form •(r) = re-ar where a is an adjustable parameter. Optimize a so that you obtain the minimum energy (i.e., find the extremum by imposing (E) = 0). How does the minimum energy compare to the ground state energy of an electron? Hint: n! for a>0 A nEN ne-andc= +1 Integration of function f(r, 0,6) in spherical coordinates: $*$*$* $(1,0, 6)r? sin ødødødr f(,0,)ra sin døddr
2. Potential Inside a Sphere We are interested in the electric potential inside a spherical shell that is radius a and centered on the origin. There are no charges inside the she, so the potential satisfies the Laplace equation, However, there is an external voltage applied to the surface of the shell which holds the potential on the surface to a value which depends on θ: As a result, the potential Ф(r,0) -by symmetry, it does not depend on ф-is...
(a) Use the variational method to estimate the ground state energy of a particle of |mass m in a potential Vx)kx, k > 0. (b) Calculate the energy shift in the ground state and in the degenerate 1t excited state of a 2-dimensional harmonic oscillator H(P2P,2/2m m(x +y)due to the perturbation V 2Axy. (20 pts)
(a) Use the variational method to estimate the ground state energy of a particle of |mass m in a potential Vx)kx, k > 0. (b)...
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