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2.3 The wave function of a harmonic oscillator with the parameters m and w is a...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x,...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x, t) = (x)e-1st/ where and E, are the harmonic oscillator's stationary states and their corresponding energies. (a) Show that the expectation value of position is (hint: use the results of Problem 4): (v) = A cos (wt - ) where the real constants A and o are given by: 1 2 Ae-id-1 " Entichtin Interpret this result, comparing it with the motion of a...
3- A one-dimensional harmonic oscillator wave function is ψ(x) = Axe-bx2 a) Find the total energy...
3- A one-dimensional harmonic oscillator wave function is ψ(x) = Axe-bx2 a) Find the total energy E b) Find the constant b c) Find the normalization constant A. d) Find the expectation value of x, e) Find the uncertainty in x, Ох. f) Find the expectation value of p g) Find the uncertainty in p, Op For the Hamiltonian matrix shown below:
3- A one-dimensional harmonic oscillator wave function is ψ(x) = Axe-bx2 a) Find the total energy E b)...
[20 points] A particle in the simple harmonic oscillator potential with angular frequency a is initially...
[20 points] A particle in the simple harmonic oscillator potential with angular frequency a is initially in the ground state: c,y, (x) =Yo(x Att = 0 , the angular frequency of the oscillator suddenly doubles: a} → a½-2.4 The initial wave function can be written in terms of the modified potential (denoted with a tilde:~: Recall that the general form of the first few stationary states for the harmonic oscillator are given on page 56 of your text. a. What...
For a particle described as a harmonic oscillator, the total energy w given by E,- (n...
For a particle described as a harmonic oscillator, the total energy w given by E,- (n + hy and the potential energy is piven by VG) kw The classical turning points, to are the values of x where the total energy is equal to the potential energy. The ground state wave function of a harc oscillator is . The cost is defined by a = k/?. If we define the variable y as y = x, which of the following...
3) A particle of mass m in the harmonic oscillator potential is initially described by a...
3) A particle of mass m in the harmonic oscillator potential is initially described by a wave function with B- where A is a normalization constant a) What is the expectation value of the energy? b) Write Y[xt). [Hint: First write Ψas a linear superposition of SHO eigenfunctions, un(A) Try to figure out a way to do it without evaluating the overlap integrals.]
The three-dimensional harmonic oscillator Cartesian wave functions that you found in Prob. 4.46 a...
The three-dimensional harmonic oscillator Cartesian wave functions that you found in Prob. 4.46 are simultaneous eigenfunctions of H and parity (i.e., r →-r), but they are not also simultaneous eigenfunctions of L' and Lz. However, we know that it's possible to construct eigenfunctions of H for the 3D harmonic oscillator that are also eigenfunctions of L, Lz, and parity. Combine the Gaussian factors that appear in your Prob. 4.46 eigenfunctions into a function of r that is independent of θ...
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...
1) Wave function for the ground state of an harmonic oscillator is given by. (x) =...
1) Wave function for the ground state of an harmonic oscillator is given by. (x) = A1/2 (a/T)1/4 e-ax /2 Evaluate the expectation value <x<> for this wave state (ove (Hint: Joo.co u² e-a u du = 2;. ue-au du = (1/2a) (Tc/a)2) pace)
2. Consider a one-dimensional simple harmonic oscillator. Do the following algebraically. 2. Consider a one -dimensional...
2. Consider a one-dimensional simple harmonic oscillator. Do the
following algebraically.
2. Consider a one -dimensional simple harmonic oscillator. Do the following algebraically. a) Construct a linear superposition of I0) and |1) by choosing appropriate phases, such that (x) is as large as possible. i) Suppose the oscillator is in the state la) att 0. Find la) and x) ii) Evaluate the expectation value(x) at t 〉 0. ii Evaluate (A x)2) att > 0. b) Answer the same questions...
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...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x, t) = (x)e-1st/ where and E, are the harmonic oscillator's stationary states and their corresponding energies. (a) Show that the expectation value of position is (hint: use the results of Problem 4): (v) = A cos (wt - ) where the real constants A and o are given by: 1 2 Ae-id-1 " Entichtin Interpret this result, comparing it with the motion of a...
3- A one-dimensional harmonic oscillator wave function is ψ(x) = Axe-bx2 a) Find the total energy E b) Find the constant b c) Find the normalization constant A. d) Find the expectation value of x, e) Find the uncertainty in x, Ох. f) Find the expectation value of p g) Find the uncertainty in p, Op For the Hamiltonian matrix shown below:
3- A one-dimensional harmonic oscillator wave function is ψ(x) = Axe-bx2 a) Find the total energy E b)...
[20 points] A particle in the simple harmonic oscillator potential with angular frequency a is initially in the ground state: c,y, (x) =Yo(x Att = 0 , the angular frequency of the oscillator suddenly doubles: a} → a½-2.4 The initial wave function can be written in terms of the modified potential (denoted with a tilde:~: Recall that the general form of the first few stationary states for the harmonic oscillator are given on page 56 of your text. a. What...
For a particle described as a harmonic oscillator, the total energy w given by E,- (n + hy and the potential energy is piven by VG) kw The classical turning points, to are the values of x where the total energy is equal to the potential energy. The ground state wave function of a harc oscillator is . The cost is defined by a = k/?. If we define the variable y as y = x, which of the following...
3) A particle of mass m in the harmonic oscillator potential is initially described by a wave function with B- where A is a normalization constant a) What is the expectation value of the energy? b) Write Y[xt). [Hint: First write Ψas a linear superposition of SHO eigenfunctions, un(A) Try to figure out a way to do it without evaluating the overlap integrals.]
The three-dimensional harmonic oscillator Cartesian wave functions that you found in Prob. 4.46 are simultaneous eigenfunctions of H and parity (i.e., r →-r), but they are not also simultaneous eigenfunctions of L' and Lz. However, we know that it's possible to construct eigenfunctions of H for the 3D harmonic oscillator that are also eigenfunctions of L, Lz, and parity. Combine the Gaussian factors that appear in your Prob. 4.46 eigenfunctions into a function of r that is independent of θ...
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...
1) Wave function for the ground state of an harmonic oscillator is given by. (x) = A1/2 (a/T)1/4 e-ax /2 Evaluate the expectation value <x<> for this wave state (ove (Hint: Joo.co u² e-a u du = 2;. ue-au du = (1/2a) (Tc/a)2) pace)
2. Consider a one-dimensional simple harmonic oscillator. Do the
following algebraically.
2. Consider a one -dimensional simple harmonic oscillator. Do the following algebraically. a) Construct a linear superposition of I0) and |1) by choosing appropriate phases, such that (x) is as large as possible. i) Suppose the oscillator is in the state la) att 0. Find la) and x) ii) Evaluate the expectation value(x) at t 〉 0. ii Evaluate (A x)2) att > 0. b) Answer the same questions...
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...
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