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The wave function of a particle with mass m2 at t= 0 has the following form:...
3. A particle of mass m in a one-dimensional box has the following wave function in...
3. A particle of mass m in a one-dimensional box has the following wave function in the region x-0 tox-L: ? (x.r)=?,(x)e-iEy /A +?,(X)--iE//h Here Y,(x) and Y,(x) are the normalized stationary-state wave functions for the n = 1 and n = 3 levels, and E1 and E3 are the energies of these levels. The wave function is zero for x< 0 and forx> L. (a) Find the value of the probability distribution function atx- L/2 as a function of...
5. A particle in the harmonic oscillator potential has the initial wave function Psi(x, 0) = A[\psi_{0}(x) + \psi_{1}(x)] for some constant A. Here to and ₁ are the normalized ground state and the first excited state wavefunctions of the harmonic oscill
5. A particle in the harmonic oscillator potential has the initial wave function Psi(x, 0) = A[\psi_{0}(x) + \psi_{1}(x)] for some constant A. Here to and ₁ are the normalized ground state and the first excited state wavefunctions of the harmonic oscillator, respectively. (a) Normalize (r, 0). (b) Find the wavefunction (r, t) at a later time t and hence evaluate (x, t) 2. Leave your answers involving expressions in to and ₁. c) sing the following normalized expression of...
3. At time t-0 a particle is represented by the wave function A-if 0 < x<a...
3. At time t-0 a particle is represented by the wave function A-if 0 < x<a ψ(x,0) = 0 otherwise where A, a, and b are constants. a) Normalize ψ(x,0). b) Draw (x,0). c) Where is the particle most likely to be found at t-0? d) What is the probability of finding the particle to the left of a? e) What is the expectation value of x?
h2 4. In a region of the x-axis, a particle has a wave function given by...
h2 4. In a region of the x-axis, a particle has a wave function given by y(x) = Ae-*4722° and energy where L is some length. (a) Find the potential energy as a function of x, and sketch V (x) versus x. (b) What is the classical potential (or corresponding force function) that has this dependence? (c) Find the kinetic energy as a function of x. (d) Show that x = L is the classical turning point (i.e. the place...
Q3) A particle in the harmonic oscillator potential has the initial normalized wave function Ψ(?, 0)...
Q3) A particle in the harmonic oscillator potential has the initial normalized wave function Ψ(?, 0) = 1 /√5 [2 ?₁ (?) + ?₂ (?)] where ?1 and ?2 are the eigenfunctions of the oscillator Hamiltonian for ? = 1,2 states. a) Write down the expression for Ψ(?,?). b) Calculate the probability density ℙ(?,?) = |Ψ(?,?)| ² . Express it as a sinusoidal function of time. To simplify the result, let ? ≡ (?² ℏ)/ 2??² . c) Calculate 〈?〉...
In a one-dimensional system at time t-0, the wave function of a particle is given by...
In a one-dimensional system at time t-0, the wave function of a particle is given by the function xfor 0SxSL 0 elsewhere -A opl as sketched in the diagram, where A is a positive constant. If the position of the particle is measured at time t-0, what is the probability of finding it somewhere in the interval 0 sx S L22 Specify your answer as a fraction or as a decimal correct to 2 significant figures. probability
2. A particle of mass m in the infinite square well of width a (located at...
2. A particle of mass m in the infinite square well of width a (located at 0 SSa) has as its initial wave function a mixture of two stationary states: v(x,0)Avi(x) +2s (x). (a) Find the probability density of finding the particle at the center of the well, as a function of time. (b) Find the average momentum of the particle at time t.
Consider a particle of mass m that is described by the wave function (x, t) =...
Consider a particle of mass m that is described by the wave function (x, t) = Ce~iwte-(x/l)2 where C and l are real and positive constants, with / being the characteristic length-scale in the problem Calculate the expectation values of position values of 2 and p2. and momentum p, as well as the expectation Find the standard deviations O and op. Are they consistent with the uncertainty principle? to be independent What should be the form of the potential energy...
Help with question (e) 3. Wave functions. For this question, suppose a particle of mass m...
Help with question (e)
3. Wave functions. For this question, suppose a particle of mass m is described by the wave function 23ha Axe-ax2c"Tat Ψ(x, t) (a) Show that the wave function above is a solution to the full time dependent Schrödinger equation if the potential energy function is given by 2h2a2 U(x)2 (b) What are the dimensional units of a and A in the wave function? (c) Show that the probability distribution associated with V(r, r) is independent of...
0 is given by a gaussian wave packet Consider a free particle whose state at time...
0 is given by a gaussian wave packet Consider a free particle whose state at time t (x, 0) Ae2/a2 for real constants A, a. (a) Normalize (r, 0), i.e., find A (b) Find (r, t). You can do the integral by completing the square in the exponent to get it into the form of a gaussian (c) Compute the probability density (, t), expressing your answer in terms of the quantity w av1(2ht/ma2)2 Sketch the probability density as a...
3. A particle of mass m in a one-dimensional box has the following wave function in the region x-0 tox-L: ? (x.r)=?,(x)e-iEy /A +?,(X)--iE//h Here Y,(x) and Y,(x) are the normalized stationary-state wave functions for the n = 1 and n = 3 levels, and E1 and E3 are the energies of these levels. The wave function is zero for x< 0 and forx> L. (a) Find the value of the probability distribution function atx- L/2 as a function of...
3. At time t-0 a particle is represented by the wave function A-if 0 < x<a ψ(x,0) = 0 otherwise where A, a, and b are constants. a) Normalize ψ(x,0). b) Draw (x,0). c) Where is the particle most likely to be found at t-0? d) What is the probability of finding the particle to the left of a? e) What is the expectation value of x?
h2 4. In a region of the x-axis, a particle has a wave function given by y(x) = Ae-*4722° and energy where L is some length. (a) Find the potential energy as a function of x, and sketch V (x) versus x. (b) What is the classical potential (or corresponding force function) that has this dependence? (c) Find the kinetic energy as a function of x. (d) Show that x = L is the classical turning point (i.e. the place...
In a one-dimensional system at time t-0, the wave function of a particle is given by the function xfor 0SxSL 0 elsewhere -A opl as sketched in the diagram, where A is a positive constant. If the position of the particle is measured at time t-0, what is the probability of finding it somewhere in the interval 0 sx S L22 Specify your answer as a fraction or as a decimal correct to 2 significant figures. probability
2. A particle of mass m in the infinite square well of width a (located at 0 SSa) has as its initial wave function a mixture of two stationary states: v(x,0)Avi(x) +2s (x). (a) Find the probability density of finding the particle at the center of the well, as a function of time. (b) Find the average momentum of the particle at time t.
Consider a particle of mass m that is described by the wave function (x, t) = Ce~iwte-(x/l)2 where C and l are real and positive constants, with / being the characteristic length-scale in the problem Calculate the expectation values of position values of 2 and p2. and momentum p, as well as the expectation Find the standard deviations O and op. Are they consistent with the uncertainty principle? to be independent What should be the form of the potential energy...
Help with question (e)
3. Wave functions. For this question, suppose a particle of mass m is described by the wave function 23ha Axe-ax2c"Tat Ψ(x, t) (a) Show that the wave function above is a solution to the full time dependent Schrödinger equation if the potential energy function is given by 2h2a2 U(x)2 (b) What are the dimensional units of a and A in the wave function? (c) Show that the probability distribution associated with V(r, r) is independent of...
0 is given by a gaussian wave packet Consider a free particle whose state at time t (x, 0) Ae2/a2 for real constants A, a. (a) Normalize (r, 0), i.e., find A (b) Find (r, t). You can do the integral by completing the square in the exponent to get it into the form of a gaussian (c) Compute the probability density (, t), expressing your answer in terms of the quantity w av1(2ht/ma2)2 Sketch the probability density as a...
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