Convince yourself that function exp(-x2/2) is an eigenfunction of the operator (1/2)(-d2/dx2 + x2). Compute the...
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Convince yourself that function exp(-x2/2) is an eigenfunction of the operator (1/2)(-d2/dx2 + x2). Compute the corresponding eigenvalue. (We will see in class that this operator is the Hamiltonian for the harmonic oscillator, if one sets the mass, frequency, and the Planck's constant at 1.) |
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Convince yourself that function exp(-x2/2) is an eigenfunction of the operator (1/2)(-d2/dx2 + x2). Compute the...
Is the function x2e−ax^(2) an eigenfunction of operator d2/dx2 − 4a2x2. If it is then what...
Is the function x2e−ax^(2) an eigenfunction of operator d2/dx2 − 4a2x2. If it is then what is the corresponding eigenvalue?
Consider the harmonic oscillator wave function 1/4 where α = (-)"*. Here k, is the stiffness...
Consider the harmonic oscillator wave function 1/4 where α = (-)"*. Here k, is the stiffness coefficient of the oscillator and m is mass. Recall that the oscillation frequency iso,s:,k, / m In class we showed that Ψ0(x) Is an eigenfunction of the Hamiltonian, with an eigenvalue Eo (1/2)ha a) Normalize the wave function in Eq.(1) b) Graph the probability density. Note that a has units of length and measures the "width" of the wave function. It's easier to use...
a) Show that the wave function y(x) = N exp( – x²/(2a?)) with a? = ()...
a) Show that the wave function y(x) = N exp( – x²/(2a?)) with a? = () is a solution of the Schrödinger equation for harmonic oscillator with potential V(x) = k x2/2. (10 pt) b) What is the energy of harmonic oscillator with the wave function y(x) in terms of k and m? (5 pt) c) Sketch the potential energy of harmonic oscillator, the energy level corresponding to y(x), the wave function (x), and the probability density associated with y(x)...
Consider the following second order linear operator: 82 with Notice, that if instead of 3 we had 2 there, we would get a Legendre operator (whose eigenfunctions are Legendre polynomials). But no...
Consider the following second order linear operator: 82 with Notice, that if instead of 3 we had 2 there, we would get a Legendre operator (whose eigenfunctions are Legendre polynomials). But nothing can be further from it than the operator above. The eigenvalue/eigenfunction problem, emerged in the analysis of vibrations of a particular quant urn liquid. An eigenvalue λ corresponds to an excitation mode of frequency Ω = V The eigenfunction ψ(r) would give a spatial profile of the deviation...
2. [10 points] For a Simple Harmonic Oscillator: a) Draw ψ(x) and ψ2(x) for the u,...
2. [10 points] For a Simple Harmonic Oscillator: a) Draw ψ(x) and ψ2(x) for the u, and v-6 states. Make sure to include as much detail as possible. b) Show that 2(3) is normalized. 3. [5] A certain non-halogen diatomic molecule was found to have a force constant of 99 N/m and an observed vibrational frequency of 162.2 cm1 Determine the identity of the unknown diatomic 4. [10 points) a) What is the difference between commuting and non-commuting operators? What...
a) Discuss why the de Broglie wavelength λ corresponding to a momentum p (p wavenumber given...
a) Discuss why the de Broglie wavelength λ corresponding to a momentum p (p wavenumber given by k # 2n/A) leads to a representation of p by the operator p as (h/) (d/dx) hk, where k is the b) Using theoperao orm of p given in part a, show that,pih c) The total energy of a simple harmonic oscillator of mass M and spring constant K can be written as H- p2/M + ke . If the mass is displaced...
Consider the finite difference matrix operator for the 1D model problem u(/d2- f(x) on domain [0,...
Consider the finite difference matrix operator for the 1D model problem u(/d2- f(x) on domain [0, 1] with boundary conditions u(0) = 0 and u(1) = 0, given by [-2 1 1-2 1 E RnXn h2 1 -2 1 This matrix can be considered a discrete version of the continuous operator d/da2 that acts upon a function(r). (a) Show that the n eigenvectors of A are given by the vectors ) (p-1,... , n) with components and with eigenvalues h2...
Q10 The Hamiltonian of a two-state system is given by H E ( i)- I02)(2 |...
Q10 The Hamiltonian of a two-state system is given by H E ( i)- I02)(2 | -i | ¢1)(2 | +i | ¢2) (¢1 1) where , p2) form a complete and orthonormal basis; E is a real constant having the dimensions of energy (a) Is H Hermitian? Calculate the trace of H (b) Find the matrix representing H in the | øı), | 42) basis and calculate the eigenvalues and the eigenvectors of the matrix. Calculate the trace of...
Two students have a very pressing homework deadline concerning the application of the variational principle to...
Two students have a very pressing homework deadline concerning the application of the variational principle to estimate the ground state energy of the harmonic oscillator. The Hamiltonian operator of such system is î H -12d = 24 d.22 + 2 .2. in which u is the reduced mass of the oscillator and w = (force constant/u)/2 its natural frequency. The correct energies for this system are well known Eo = (v +) , v= 0,1,2, ... As the trial function...
Please answer number 8 l Verizon LTE 9:53 PM 100%,--+ Close Physical Chemistry ll Spring...1 DOCX-149...
Please answer number 8
l Verizon LTE 9:53 PM 100%,--+ Close Physical Chemistry ll Spring...1 DOCX-149 KB (e) none of the above 7. A free particle is inside a one dimentional box from 0 to a/2, (a is a constant). If the particle is in the first excited states with eigenfunction, y Nsin (4px/a) (a) Determine the normalization constant. (b) Calculate the probability in between a/4 and a/2 8. What is the degree of the degeneracy if the three quantum...
Consider the harmonic oscillator wave function 1/4 where α = (-)"*. Here k, is the stiffness coefficient of the oscillator and m is mass. Recall that the oscillation frequency iso,s:,k, / m In class we showed that Ψ0(x) Is an eigenfunction of the Hamiltonian, with an eigenvalue Eo (1/2)ha a) Normalize the wave function in Eq.(1) b) Graph the probability density. Note that a has units of length and measures the "width" of the wave function. It's easier to use...
a) Show that the wave function y(x) = N exp( – x²/(2a?)) with a? = () is a solution of the Schrödinger equation for harmonic oscillator with potential V(x) = k x2/2. (10 pt) b) What is the energy of harmonic oscillator with the wave function y(x) in terms of k and m? (5 pt) c) Sketch the potential energy of harmonic oscillator, the energy level corresponding to y(x), the wave function (x), and the probability density associated with y(x)...
Consider the following second order linear operator: 82 with Notice, that if instead of 3 we had 2 there, we would get a Legendre operator (whose eigenfunctions are Legendre polynomials). But nothing can be further from it than the operator above. The eigenvalue/eigenfunction problem, emerged in the analysis of vibrations of a particular quant urn liquid. An eigenvalue λ corresponds to an excitation mode of frequency Ω = V The eigenfunction ψ(r) would give a spatial profile of the deviation...
2. [10 points] For a Simple Harmonic Oscillator: a) Draw ψ(x) and ψ2(x) for the u, and v-6 states. Make sure to include as much detail as possible. b) Show that 2(3) is normalized. 3. [5] A certain non-halogen diatomic molecule was found to have a force constant of 99 N/m and an observed vibrational frequency of 162.2 cm1 Determine the identity of the unknown diatomic 4. [10 points) a) What is the difference between commuting and non-commuting operators? What...
a) Discuss why the de Broglie wavelength λ corresponding to a momentum p (p wavenumber given by k # 2n/A) leads to a representation of p by the operator p as (h/) (d/dx) hk, where k is the b) Using theoperao orm of p given in part a, show that,pih c) The total energy of a simple harmonic oscillator of mass M and spring constant K can be written as H- p2/M + ke . If the mass is displaced...
Consider the finite difference matrix operator for the 1D model problem u(/d2- f(x) on domain [0, 1] with boundary conditions u(0) = 0 and u(1) = 0, given by [-2 1 1-2 1 E RnXn h2 1 -2 1 This matrix can be considered a discrete version of the continuous operator d/da2 that acts upon a function(r). (a) Show that the n eigenvectors of A are given by the vectors ) (p-1,... , n) with components and with eigenvalues h2...
Q10 The Hamiltonian of a two-state system is given by H E ( i)- I02)(2 | -i | ¢1)(2 | +i | ¢2) (¢1 1) where , p2) form a complete and orthonormal basis; E is a real constant having the dimensions of energy (a) Is H Hermitian? Calculate the trace of H (b) Find the matrix representing H in the | øı), | 42) basis and calculate the eigenvalues and the eigenvectors of the matrix. Calculate the trace of...
Two students have a very pressing homework deadline concerning the application of the variational principle to estimate the ground state energy of the harmonic oscillator. The Hamiltonian operator of such system is î H -12d = 24 d.22 + 2 .2. in which u is the reduced mass of the oscillator and w = (force constant/u)/2 its natural frequency. The correct energies for this system are well known Eo = (v +) , v= 0,1,2, ... As the trial function...
Please answer number 8
l Verizon LTE 9:53 PM 100%,--+ Close Physical Chemistry ll Spring...1 DOCX-149 KB (e) none of the above 7. A free particle is inside a one dimentional box from 0 to a/2, (a is a constant). If the particle is in the first excited states with eigenfunction, y Nsin (4px/a) (a) Determine the normalization constant. (b) Calculate the probability in between a/4 and a/2 8. What is the degree of the degeneracy if the three quantum...
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