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4. (20 points). Consider a quantum harmonic oscillator with characteristic frequency w. The syste...
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...
3. The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an...
3. The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Consider an electron trapped by a one-dimensional harmonic potential V(x)=-5 mo?x” (where m is the electron mass, o is a constant angular frequency). In this case, the Schrödinger equation takes the following form, **...
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...
1. The energy levels of a quantum harmonic oscillator are given by E Planck constant, w...
1. The energy levels of a quantum harmonic oscillator are given by E Planck constant, w is the frequency of oscillation and n-0,1,2, Determine the following: (a) Show that the one-particle partition function is given by 211-exp-Bhu) oan 1 (1 Hint you will need to use the following formula for a geometric progression: (b) Show that the internal energy is given by (c) Show that the Helmoltz free energy is given by 1 In(1 exp Bha) (d) Show that the...
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...
3. The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Consider an electron trapped by a one-dimensional harmonic potential V(x)=-5 mo?x” (where m is the electron mass, o is a constant angular frequency). In this case, the Schrödinger equation takes the following form, **...
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...
1. The energy levels of a quantum harmonic oscillator are given by E Planck constant, w is the frequency of oscillation and n-0,1,2, Determine the following: (a) Show that the one-particle partition function is given by 211-exp-Bhu) oan 1 (1 Hint you will need to use the following formula for a geometric progression: (b) Show that the internal energy is given by (c) Show that the Helmoltz free energy is given by 1 In(1 exp Bha) (d) Show that the...
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