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Problem #7 The quantum harmonic oscillator Hamiltonian, expressed in terms of raising and lowering operators, is We also know that Using these two statements, show that if then both Yn and ah have we...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian of a particle of a simple Harmonic oscillator potential in one dimension can be expressed in term of the creation and annihilation operators àt and à, respectively, as: or with In >, n = 0,1,..) are the nth eigenstates of the above Hamiltonian. Part A A.1. Show that the energy levels of a simple harmonic oscillator are E,' Aw (nti), n=0, 12, A.2. Calculate...
4. (30 points) Harmonic oscillator with perturbation Recall the Hamiltonian of an harmonic oscillator in 1D:...
4. (30 points) Harmonic oscillator with perturbation Recall the Hamiltonian of an harmonic oscillator in 1D: p21 ÃO = + mwf?, where m is the mass of the particle and w is the angular frequency. Now, let us perturb the oscillator with a quadratic potential. The perturbation is given by Î' = zgmw?h?, where g is a dimensionless constant and g <1. (a) Write down the eigen-energies of the unperturbed Hamiltonian. (b) In Lecture 3, we introduced the lowering (or...
6 The Fermionic Oscillator Suppose that we constructed a harmonic oscillator Hamiltonian H in terms of...
6 The Fermionic Oscillator Suppose that we constructed a harmonic oscillator Hamiltonian H in terms of raising and lowering operators a+,a in the usual way, such that but now whereaa obey the anticommutation relationn (Be careful! The a+,a are operators, rather than numbers.) (a) Suppose I give you a wavefunction that solves the time-independent Schrödinger equation, i.e. such that HUn-EUn-hw (n + ) ψη. Is a+Un also a solution to the time-independent Schrödinger equation If so, what is its energy...
Intro to Quantum Mechanics problem: . In a harmonic oscillator a normalized "coherent" state ya(x) is...
Intro to Quantum Mechanics problem:
. In a harmonic oscillator a normalized "coherent" state ya(x) is defined in terms of the lowering operator a. by aXa(x) = a Xa(x) for some (complex) number a. /Coherent states have many applications in atomic, molecular, and optical physics, for instance lasers and Bose-Einstein condensates]. (a) Using the properties for any wavefunctions f(x) and g(x) that 00 00 if ag dx (a.f)g dx f a+g dx (a.)'g dx -00 -00 -00 calculate <x >...
7 Harmonic oscillator in "energy space" Consider the harmonic oscillator in "energy space", i.e., in terms...
7 Harmonic oscillator in "energy space" Consider the harmonic oscillator in "energy space", i.e., in terms of the basis of eigenvectors n) of the harmonic oscillator Hamiltonian, with Hn) -hwn1/2)]n). We computed these in terms of wavefunctions in position space, ie. pn(x)-(zln), but we can also work purely in terms of the abstract energy eigenvectors in Dirac notation. PS9.pdf 1. You computed the matrix elements 〈nleln) on an earlier problem set. Now find (nn) for general n,n' 2. Find the...
In class we solved the quantum harmonic oscillator problem for a diatomic molecule. As part of...
In class we solved the quantum harmonic oscillator problem for a diatomic molecule. As part of that solution we transformed coordinates from x, the oscillator displacement coordinate, to the unitless, y using the relationship where μ is the reduced mass of the diatomic molecule and k is the force constant. The solutions turned out to be: w(y)N,H, (y)e Where N is a normalization constant, H,(v) are the Hermit polynomials and v is the quantum number with values of v0,1,2,3,.. The...
quantum mechanic II d show that d5 8) Using a Age^ 7 is a solution to...
quantum mechanic II
d show that d5 8) Using a Age^ 7 is a solution to the ground state condition: a y = 0. The problems in this Assignment make use of the following: The ladder operators, defined in terms of the dimensionless position () variable ax, where a as: d 1 and The number operator: N = â'à The Harmonic Oscillator Hamiltonian: ) тод d2 2 d2 m 2m 2 2
8.4 The Two-Dimensional Central-Force Problem The 2D harmonic oscillator is a 2D central force pr...
8.4 The Two-Dimensional Central-Force Problem The 2D harmonic oscillator is a 2D central force problem (as discussed in TZD Many physical systems involve a particle that moves under the influence of a central force; that is, a force that always points exactly toward, or away from, a force center O. In classical mechanics a famous example of a central force is the force of the sun on a planet. In atomic physics the most obvious example is the hydrogen atom,...
7. We know that well-defined preferences over two goods have the properties that () indifference curves...
7. We know that well-defined preferences over two goods have the properties that () indifference curves are negatively sloped, and (ii) that indifference curves are convex (so that chords between two points on the indifference curve lie in the set ((c, y) such that (c,y)(coyo) when (co. o) is a point on the indifference curve). Suppose that good y is "clean air" and good c is consumption of all other goods. This problem gets you to determine what these two...
constraints, also written in terms of decision variables (some using both decision supplement. In addition, the...
constraints, also written in terms of decision variables (some using both decision supplement. In addition, the Lecture notes page in this website gives you access Then you complete Task 1, you should have two decision variables, the ng constraints. We simply list all conditions that must objective function written in terms of those decision variables, and five 2. The second step is to find a solution. Task 2 tells you specifically to use Excel Solver to find this solution. The...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian of a particle of a simple Harmonic oscillator potential in one dimension can be expressed in term of the creation and annihilation operators àt and à, respectively, as: or with In >, n = 0,1,..) are the nth eigenstates of the above Hamiltonian. Part A A.1. Show that the energy levels of a simple harmonic oscillator are E,' Aw (nti), n=0, 12, A.2. Calculate...
4. (30 points) Harmonic oscillator with perturbation Recall the Hamiltonian of an harmonic oscillator in 1D: p21 ÃO = + mwf?, where m is the mass of the particle and w is the angular frequency. Now, let us perturb the oscillator with a quadratic potential. The perturbation is given by Î' = zgmw?h?, where g is a dimensionless constant and g <1. (a) Write down the eigen-energies of the unperturbed Hamiltonian. (b) In Lecture 3, we introduced the lowering (or...
6 The Fermionic Oscillator Suppose that we constructed a harmonic oscillator Hamiltonian H in terms of raising and lowering operators a+,a in the usual way, such that but now whereaa obey the anticommutation relationn (Be careful! The a+,a are operators, rather than numbers.) (a) Suppose I give you a wavefunction that solves the time-independent Schrödinger equation, i.e. such that HUn-EUn-hw (n + ) ψη. Is a+Un also a solution to the time-independent Schrödinger equation If so, what is its energy...
Intro to Quantum Mechanics problem:
. In a harmonic oscillator a normalized "coherent" state ya(x) is defined in terms of the lowering operator a. by aXa(x) = a Xa(x) for some (complex) number a. /Coherent states have many applications in atomic, molecular, and optical physics, for instance lasers and Bose-Einstein condensates]. (a) Using the properties for any wavefunctions f(x) and g(x) that 00 00 if ag dx (a.f)g dx f a+g dx (a.)'g dx -00 -00 -00 calculate <x >...
7 Harmonic oscillator in "energy space" Consider the harmonic oscillator in "energy space", i.e., in terms of the basis of eigenvectors n) of the harmonic oscillator Hamiltonian, with Hn) -hwn1/2)]n). We computed these in terms of wavefunctions in position space, ie. pn(x)-(zln), but we can also work purely in terms of the abstract energy eigenvectors in Dirac notation. PS9.pdf 1. You computed the matrix elements 〈nleln) on an earlier problem set. Now find (nn) for general n,n' 2. Find the...
In class we solved the quantum harmonic oscillator problem for a diatomic molecule. As part of that solution we transformed coordinates from x, the oscillator displacement coordinate, to the unitless, y using the relationship where μ is the reduced mass of the diatomic molecule and k is the force constant. The solutions turned out to be: w(y)N,H, (y)e Where N is a normalization constant, H,(v) are the Hermit polynomials and v is the quantum number with values of v0,1,2,3,.. The...
quantum mechanic II
d show that d5 8) Using a Age^ 7 is a solution to the ground state condition: a y = 0. The problems in this Assignment make use of the following: The ladder operators, defined in terms of the dimensionless position () variable ax, where a as: d 1 and The number operator: N = â'à The Harmonic Oscillator Hamiltonian: ) тод d2 2 d2 m 2m 2 2
8.4 The Two-Dimensional Central-Force Problem The 2D harmonic oscillator is a 2D central force problem (as discussed in TZD Many physical systems involve a particle that moves under the influence of a central force; that is, a force that always points exactly toward, or away from, a force center O. In classical mechanics a famous example of a central force is the force of the sun on a planet. In atomic physics the most obvious example is the hydrogen atom,...
7. We know that well-defined preferences over two goods have the properties that () indifference curves are negatively sloped, and (ii) that indifference curves are convex (so that chords between two points on the indifference curve lie in the set ((c, y) such that (c,y)(coyo) when (co. o) is a point on the indifference curve). Suppose that good y is "clean air" and good c is consumption of all other goods. This problem gets you to determine what these two...
constraints, also written in terms of decision variables (some using both decision supplement. In addition, the Lecture notes page in this website gives you access Then you complete Task 1, you should have two decision variables, the ng constraints. We simply list all conditions that must objective function written in terms of those decision variables, and five 2. The second step is to find a solution. Task 2 tells you specifically to use Excel Solver to find this solution. The...
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