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4 Spherical harmonics in action Consider a particle in three dimensions in a state (2, y,...
4.) A spherical square well is a square well in three spatial dimensions which satisfies -Vo,...
4.) A spherical square well is a square well in three spatial dimensions which satisfies -Vo, ifr <a/2; (3) V (r) = 0, ifr>a/2, where r is the radial coordinate and a is a constant. a) Determine the ground-state solution to the effectively one-dimensional Schroedinger equation in r for a particle of mass m, as well as the transcendental equation the ground-state energy must satisfy. b) If you replace r by a, the one-dimensional coordinate, does your answer differ from...
PROBLEMS 2.32 The wavefunction for a particle in one dimension is given by Another state that...
PROBLEMS 2.32 The wavefunction for a particle in one dimension is given by Another state that the particle may be in is A third state the particle may be in is y2/4 Normalize all three states in the interval-oo < y <-co (i.e., find A1,A2, and A3) is the probability of finding the particle in the interval 0 y < 1 when the particle : is in the state vs the same as the sum of the separate probabilities for...
Consider a particle confined to one dimension and positive with the wave function Nxear, x20 x<0...
Consider a particle confined to one dimension and positive with the wave function Nxear, x20 x<0 0 where N is a real normalization constant and α is a real positive constant with units of (length)-1. For the following, express your answers in terms of α: a) Find the normalization constant N. What are the units of your result and do they make sense? b) What is the most probable location to find the particle, or more precisely, at what z...
Consider a particle confined to one dimension and positive r with the wave function 0, z<0...
Consider a particle confined to one dimension and positive r with the wave function 0, z<0 where N is a real normalization constant and o is a real positive constant with units of (length)-1. For the following, express your answers in terms of a: a) Calculate the momentum space wave function. b) Verify that the momentum space wave function is normalized such that (2.4) c) Use the momentum space wave function to calculate the expectation value (p) via (2.5)
2 BALL AND PLANE Consider a spherical shell of radius R and charge per unit area...
2 BALL AND PLANE Consider a spherical shell of radius R and charge per unit area σ1 sitting at the origin. There is also an infinie plane parallel to the x- y plane sitting at z-zo with charge per unit area Oz. We will take Zo > R. Compute the electric field at the following locations: 2.1 10 POINTS The origin. 2.2 15 POINTS The point (xo,0,0) with xo> R 2.3 15 POINTS The point (X1, 0,21) with 0 <...
Question 2: A particle of mass m moves in a potential energy U(x) that is zero...
Question 2: A particle of mass m moves in a potential energy U(x) that is zero forェ* 0 and is-oo at r-0. This is am attractive delta function, very odd. Do not worry about the physical meaning of the potential, just roll with it for now. The system is described by the wave function Afor <0 where a is a real, positive constant with dimensions of 1/Length, and A is the normalization constant, treat it as a unknown complex-number for...
1. A particle of mass m moves in three dimensions, and has position r(t)-(x(t), y(t), z(t)) at ti...
Mechanics. Need help with c) and d)
1. A particle of mass m moves in three dimensions, and has position r(t)-(x(t), y(t), z(t)) at time t. The particle has potential energy V(x, y, 2) so that its Lagrangian is given by where i d/dt, dy/dt, dz/dt (a) Writing q(q2.93)-(r, y, z) and denoting by p (p,P2, ps) their associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy) H(q,p)H(g1, 92,9q3,...
1 From Wavefunction to Bra-Ket In bra-ket notation, a state y(x) is written as a ket:...
1 From Wavefunction to Bra-Ket In bra-ket notation, a state y(x) is written as a ket: 14) + (2). The inner product between two states 41(2), 42(2) is written as a bra-ket: (441\42) = |dx (z)* #2(a). If a state is a complex linear combination |V) = a1 (41) + a2 (42), then its corresponding bra is (V= a1 (01| +a(42 In this problem, we will use the simple harmonic oscillator as a concrete example. The energy eigenstates of the...
2.1 2.2 2.3 2 BALL AND PLANE Consider a spherical shell of radius R and charge...
2.1 2.2 2.3
2 BALL AND PLANE Consider a spherical shell of radius R and charge per unit area ơi sitting at the origin. There is also an infinite plane parallel to the x - y plane sitting at zzo with charge per unit area σ2. We will take z02R. Compute the electric field at the following locations: 2.1 10 POINTS The origin. 2.2 15 POINTS The point (xo.0,0) with xo>R 2.3 15 POINTS The point (x1,0, z) with 0...
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...
4.) A spherical square well is a square well in three spatial dimensions which satisfies -Vo, ifr <a/2; (3) V (r) = 0, ifr>a/2, where r is the radial coordinate and a is a constant. a) Determine the ground-state solution to the effectively one-dimensional Schroedinger equation in r for a particle of mass m, as well as the transcendental equation the ground-state energy must satisfy. b) If you replace r by a, the one-dimensional coordinate, does your answer differ from...
PROBLEMS 2.32 The wavefunction for a particle in one dimension is given by Another state that the particle may be in is A third state the particle may be in is y2/4 Normalize all three states in the interval-oo < y <-co (i.e., find A1,A2, and A3) is the probability of finding the particle in the interval 0 y < 1 when the particle : is in the state vs the same as the sum of the separate probabilities for...
Consider a particle confined to one dimension and positive with the wave function Nxear, x20 x<0 0 where N is a real normalization constant and α is a real positive constant with units of (length)-1. For the following, express your answers in terms of α: a) Find the normalization constant N. What are the units of your result and do they make sense? b) What is the most probable location to find the particle, or more precisely, at what z...
Consider a particle confined to one dimension and positive r with the wave function 0, z<0 where N is a real normalization constant and o is a real positive constant with units of (length)-1. For the following, express your answers in terms of a: a) Calculate the momentum space wave function. b) Verify that the momentum space wave function is normalized such that (2.4) c) Use the momentum space wave function to calculate the expectation value (p) via (2.5)
2 BALL AND PLANE Consider a spherical shell of radius R and charge per unit area σ1 sitting at the origin. There is also an infinie plane parallel to the x- y plane sitting at z-zo with charge per unit area Oz. We will take Zo > R. Compute the electric field at the following locations: 2.1 10 POINTS The origin. 2.2 15 POINTS The point (xo,0,0) with xo> R 2.3 15 POINTS The point (X1, 0,21) with 0 <...
Question 2: A particle of mass m moves in a potential energy U(x) that is zero forェ* 0 and is-oo at r-0. This is am attractive delta function, very odd. Do not worry about the physical meaning of the potential, just roll with it for now. The system is described by the wave function Afor <0 where a is a real, positive constant with dimensions of 1/Length, and A is the normalization constant, treat it as a unknown complex-number for...
Mechanics. Need help with c) and d)
1. A particle of mass m moves in three dimensions, and has position r(t)-(x(t), y(t), z(t)) at time t. The particle has potential energy V(x, y, 2) so that its Lagrangian is given by where i d/dt, dy/dt, dz/dt (a) Writing q(q2.93)-(r, y, z) and denoting by p (p,P2, ps) their associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy) H(q,p)H(g1, 92,9q3,...
1 From Wavefunction to Bra-Ket In bra-ket notation, a state y(x) is written as a ket: 14) + (2). The inner product between two states 41(2), 42(2) is written as a bra-ket: (441\42) = |dx (z)* #2(a). If a state is a complex linear combination |V) = a1 (41) + a2 (42), then its corresponding bra is (V= a1 (01| +a(42 In this problem, we will use the simple harmonic oscillator as a concrete example. The energy eigenstates of the...
2.1 2.2 2.3
2 BALL AND PLANE Consider a spherical shell of radius R and charge per unit area ơi sitting at the origin. There is also an infinite plane parallel to the x - y plane sitting at zzo with charge per unit area σ2. We will take z02R. Compute the electric field at the following locations: 2.1 10 POINTS The origin. 2.2 15 POINTS The point (xo.0,0) with xo>R 2.3 15 POINTS The point (x1,0, z) with 0...
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...
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