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11 Consider the one-dimensional oscillator defined on page 6. At time t function (r, t is...
A particle with mass m is in a one-dimensional simple harmonic oscillator potential. At time t = 0 it is described by the state where lo and l) are normalised energy eigenfunctions corresponding to e...
A particle with mass m is in a one-dimensional simple harmonic oscillator potential. At time t = 0 it is described by the state where lo and l) are normalised energy eigenfunctions corresponding to energies E and Ey and b and c are real constants. (a) Find b and c so that (x) is as large as possible. b) Write down the wavefunction of this particle at a time t later c)Caleulate (x) for the particle at time t (d)...
Question A2: Coherent states of the harmonic oscillator Consider a one-dimensional harmonic oscillator with the Hamiltonian...
Question A2: Coherent states of the harmonic oscillator Consider a one-dimensional harmonic oscillator with the Hamiltonian 12 12 m2 H = -2m d. 2+ 2 Here m and w are the mass and frequency, respectively. Consider a time-dependent wave function of the form <(x,t) = C'exp (-a(x – 9(t)+ ik(t)z +io(t)), where a and C are positive constants, and g(t), k(t), and o(t) are real functions of time t. 1. Express C in terms of a. [2 marks] 2. By...
At time t = 0, a mass-m particle in a one-dimensional potential well is in a...
At time t = 0, a mass-m particle in a one-dimensional potential well is in a state given by the normalised wave function (x, 0) =3/2eAl2| | -ao x << 0, realU>0. Find the potential energy V = the energy eigenvalue E. Fix zero energy according to the convention V(x) » 0 for ao. Is there a delta function singularity at x0? V (x) for which this is an energy eigenstate and determine [6]
At time t = 0, a...
3 At a given time, the normalised wave function for a particle in a one-dimensional infinite...
3 At a given time, the normalised wave function for a particle in a one-dimensional infinite square well -a < x < a is given by 2 sin2 V inside the well and zero outside. Find the probability that a measurement of energy yields the eigenvalue En. (Hint: use data on page 6.) [6] Useful Data and Formulas = 1.60 x 10-19 C Elementary charge e h/2T=1.05 x 10-34 Js Planck's constant 3.00 x 108 m s-1 Speed of light...
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...
Consider a particle of mass m under the action of the one-dimensional harmonic oscillator potential. The Hamiltonian is...
Consider a particle of mass m under the action of the one-dimensional harmonic oscillator potential. The Hamiltonian is given by Knowing that the ground state of the particle at a certain instant is described by the wave function mw 1/4 _mw2 Th / calculate (for the ground state): a) The mean value of the position <x> (2 marks) b) The mean value of the position squared < x2 > (2 marks) c) the mean value of the momentum <p> (2...
(a) At time t 0, a one-dimensional bound system is in a state described by the normalized wave function V(r,0). The sys...
(a) At time t 0, a one-dimensional bound system is in a state described by the normalized wave function V(r,0). The system has a set of orthonormal energy eigenfunctions (), 2(x),.. with corresponding eigenvalues E, E2, .... Write down the overlap rule for the probability of getting the energy E when the energy is measured at time t 0 (b) Suppose that a system is described by a normalized wave function of the form (,0) an(r), where the an are...
4) The wave functions of a one-dimensional harmonic oscillator for the states v = 0 and...
4) The wave functions of a one-dimensional harmonic oscillator for the states v = 0 and v = 1 are given by: V. (y) = Noe- 4; () = (47) 2ye and y = (Premu)/2 x Write the expression for the Hamiltonian eigenvalue equation for this system and show that yo satisfy the eigenvalue equation:
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)...
Question 3 Consider the one-dimensional harmonic oscillator, and denote its properly normalised e...
ONLY (e) (f) NEEDED THANK YOU :)
Question 3 Consider the one-dimensional harmonic oscillator, and denote its properly normalised energy eigenstates by { | n〉, n = 0, 1, 2, 3, . . .). Define the state where α is a complex number, and C is a normalisation constant. (a) Use a Campbell-Baker-Hausdorff relation (or otherwise) to show that In other words, | α > is an eigenstate of the (non-Hermitian) lowering operator with (complex) eigenvalue α. (b) During lectures...
A particle with mass m is in a one-dimensional simple harmonic oscillator potential. At time t = 0 it is described by the state where lo and l) are normalised energy eigenfunctions corresponding to energies E and Ey and b and c are real constants. (a) Find b and c so that (x) is as large as possible. b) Write down the wavefunction of this particle at a time t later c)Caleulate (x) for the particle at time t (d)...
Question A2: Coherent states of the harmonic oscillator Consider a one-dimensional harmonic oscillator with the Hamiltonian 12 12 m2 H = -2m d. 2+ 2 Here m and w are the mass and frequency, respectively. Consider a time-dependent wave function of the form <(x,t) = C'exp (-a(x – 9(t)+ ik(t)z +io(t)), where a and C are positive constants, and g(t), k(t), and o(t) are real functions of time t. 1. Express C in terms of a. [2 marks] 2. By...
At time t = 0, a mass-m particle in a one-dimensional potential well is in a state given by the normalised wave function (x, 0) =3/2eAl2| | -ao x << 0, realU>0. Find the potential energy V = the energy eigenvalue E. Fix zero energy according to the convention V(x) » 0 for ao. Is there a delta function singularity at x0? V (x) for which this is an energy eigenstate and determine [6]
At time t = 0, a...
3 At a given time, the normalised wave function for a particle in a one-dimensional infinite square well -a < x < a is given by 2 sin2 V inside the well and zero outside. Find the probability that a measurement of energy yields the eigenvalue En. (Hint: use data on page 6.) [6] Useful Data and Formulas = 1.60 x 10-19 C Elementary charge e h/2T=1.05 x 10-34 Js Planck's constant 3.00 x 108 m s-1 Speed of light...
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...
Consider a particle of mass m under the action of the one-dimensional harmonic oscillator potential. The Hamiltonian is given by Knowing that the ground state of the particle at a certain instant is described by the wave function mw 1/4 _mw2 Th / calculate (for the ground state): a) The mean value of the position <x> (2 marks) b) The mean value of the position squared < x2 > (2 marks) c) the mean value of the momentum <p> (2...
(a) At time t 0, a one-dimensional bound system is in a state described by the normalized wave function V(r,0). The system has a set of orthonormal energy eigenfunctions (), 2(x),.. with corresponding eigenvalues E, E2, .... Write down the overlap rule for the probability of getting the energy E when the energy is measured at time t 0 (b) Suppose that a system is described by a normalized wave function of the form (,0) an(r), where the an are...
4) The wave functions of a one-dimensional harmonic oscillator for the states v = 0 and v = 1 are given by: V. (y) = Noe- 4; () = (47) 2ye and y = (Premu)/2 x Write the expression for the Hamiltonian eigenvalue equation for this system and show that yo satisfy the eigenvalue equation:
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)...
ONLY (e) (f) NEEDED THANK YOU :)
Question 3 Consider the one-dimensional harmonic oscillator, and denote its properly normalised energy eigenstates by { | n〉, n = 0, 1, 2, 3, . . .). Define the state where α is a complex number, and C is a normalisation constant. (a) Use a Campbell-Baker-Hausdorff relation (or otherwise) to show that In other words, | α > is an eigenstate of the (non-Hermitian) lowering operator with (complex) eigenvalue α. (b) During lectures...
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