10. What differential equation is the one-dimensional potential equation? What is the form of the...
10. What differential equation is the one-dimensional potential equation? What is the form of the solution of the one-dimensional Dirichlet problem? The one-dimensional neumann problem?
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10. What differential equation is the one-dimensional potential equation? What is the form of the...
4. Differential equation. Show that if ψ(x) is a solution of the one-dimensional time-independent Schrödinger equation,...
4. Differential equation. Show that if ψ(x) is a solution of the one-dimensional time-independent Schrödinger equation, then c ψ(x), where c is an arbitrary complex constant, is also a solution.
Fill in the blanks (1 point) A Bernoulli differential equation is one of the form dy...
Fill in the blanks
(1 point) A Bernoulli differential equation is one of the form dy + P()y= Q(Cy" (*) dr Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u=yl-n transforms the Bernoulli equation into the linear equation du dac + (1 - n)P(x)u= (1 - nQ(2). Consider the initial value problem xy' +y= -6xy?, y(1) = -2. (a) This differential equation can be written in the form...
A particle of mass m is bound by the spherically-symmetric three-dimensional harmonic- oscillator potential energy ,...
A particle of mass m is bound by the spherically-symmetric three-dimensional harmonic- oscillator potential energy , and ф are the usual spherical coordinates. (a) In the form given above, why is it clear that the potential energy function V) is (b) For this problem, it will be more convenient to express this spherically-symmetric where r , spherically symmetric? A brief answer is sufficient. potential energy in Cartesian coordinates x, y, and z as physically the same potential energy as the...
Consider a particle of mass m moving in a one-dimensional potential of the form V. for...
Consider a particle of mass m moving in a one-dimensional potential of the form V. for 0<x<b, V(a) = 0 for Islal<e, for 1212, with V., b and c positive constants and c>b. a Explain why the wave function of the particle can be assumed to be cither an even function or an odd function of a. b For the case that the energy E of the particle is in the range 0<ESV., find the (unnormalized) even cigenfunctions and give...
Problem 5. (1 point) A Bernoulli differential equation is one of the form +P()y= Q()y" (*)...
Problem 5. (1 point) A Bernoulli differential equation is one of the form +P()y= Q()y" (*) Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u =y- transforms the Bemoulli equation into the linear equation + (1 - x)P(3)u = (1 - x)^(x). Consider the initial value problem ry' +y = -3.xy?, y(1) = 2. (a) This differential equation can be written in the form (*) with P(1) =...
Problem 1. (25 points) Consider the following differential equation. 36 (a) Using the change of variable, 2 VT, write the differential equation in the form of Bessel's equation, 22y" zy(22- v...
Problem 1. (25 points) Consider the following differential equation. 36 (a) Using the change of variable, 2 VT, write the differential equation in the form of Bessel's equation, 22y" zy(22- v2)y 0. (b) Find the general solution of the differential equation (y(). (You do not need to find the value:s of Gamma functions.) (c) Find the term multiplying ? in the solution. (You do not need to find the values of Gamma functions.)
Problem 1. (25 points) Consider the following...
A For a particle with mass m moving under a one dimensional potential V(x), one solution...
A For a particle with mass m moving under a one dimensional potential V(x), one solution to the Schrödinger equation for the region 0<x< oo is x) =2 (a>0), where A is the normalization constant. The energy of the particle in the given state is 0, Show that this function is a solution, and find the corresponding potential V(x)?
3. Consider the following differential equation 0o and a series solution to the differential equation of the form a"...
3. Consider the following differential equation 0o and a series solution to the differential equation of the form a" n-0 (a) Find the recurrence relations for the coefficients of the power series. 3 marks] (b) Determine the radius of convergence of the power series. l mar (c) Write the first eight terms of the series solution with the coefficients written in terms of ao and ai 2 marks]
3. Consider the following differential equation 0o and a series solution to...
Q1 (10 points) Consider the differential equation ty" _ y = 0. a) is this differential...
Q1 (10 points) Consider the differential equation ty" _ y = 0. a) is this differential equation linear? What is its order? Is it homogeneous? b) Try a solution of the form y=x". Is this a solution for some r? If so, find all such r. c) Based on your answer to a) about linearity and b) about what y=x" are solutions, make an educated guess a the general solution looks like. Try that guess and check that it works....
Differential equation 1. Chapter 4 covers differential equations of the form an(x)y("4a-,(x)ye-i) + +4(x)y'+...
Differential equation
1. Chapter 4 covers differential equations of the form an(x)y("4a-,(x)ye-i) + +4(x)y'+4(x)-g(x) Subject to initial conditions y)oyy-Co) Consider the second order differential equation 2x2y" + 5xy, + y-r-x 2- The Existence of a Unique Solution Theorem says there will be a unique solution y(x) to the initial-value problem at x=而over any interval 1 for which the coefficient functions, ai (x) (0 S is n) and g(x) are continuous and a, (x)0. Are there any values of x for...
4. Differential equation. Show that if ψ(x) is a solution of the one-dimensional time-independent Schrödinger equation, then c ψ(x), where c is an arbitrary complex constant, is also a solution.
Fill in the blanks
(1 point) A Bernoulli differential equation is one of the form dy + P()y= Q(Cy" (*) dr Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u=yl-n transforms the Bernoulli equation into the linear equation du dac + (1 - n)P(x)u= (1 - nQ(2). Consider the initial value problem xy' +y= -6xy?, y(1) = -2. (a) This differential equation can be written in the form...
A particle of mass m is bound by the spherically-symmetric three-dimensional harmonic- oscillator potential energy , and ф are the usual spherical coordinates. (a) In the form given above, why is it clear that the potential energy function V) is (b) For this problem, it will be more convenient to express this spherically-symmetric where r , spherically symmetric? A brief answer is sufficient. potential energy in Cartesian coordinates x, y, and z as physically the same potential energy as the...
Consider a particle of mass m moving in a one-dimensional potential of the form V. for 0<x<b, V(a) = 0 for Islal<e, for 1212, with V., b and c positive constants and c>b. a Explain why the wave function of the particle can be assumed to be cither an even function or an odd function of a. b For the case that the energy E of the particle is in the range 0<ESV., find the (unnormalized) even cigenfunctions and give...
Problem 5. (1 point) A Bernoulli differential equation is one of the form +P()y= Q()y" (*) Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u =y- transforms the Bemoulli equation into the linear equation + (1 - x)P(3)u = (1 - x)^(x). Consider the initial value problem ry' +y = -3.xy?, y(1) = 2. (a) This differential equation can be written in the form (*) with P(1) =...
Problem 1. (25 points) Consider the following differential equation. 36 (a) Using the change of variable, 2 VT, write the differential equation in the form of Bessel's equation, 22y" zy(22- v2)y 0. (b) Find the general solution of the differential equation (y(). (You do not need to find the value:s of Gamma functions.) (c) Find the term multiplying ? in the solution. (You do not need to find the values of Gamma functions.)
Problem 1. (25 points) Consider the following...
A For a particle with mass m moving under a one dimensional potential V(x), one solution to the Schrödinger equation for the region 0<x< oo is x) =2 (a>0), where A is the normalization constant. The energy of the particle in the given state is 0, Show that this function is a solution, and find the corresponding potential V(x)?
3. Consider the following differential equation 0o and a series solution to the differential equation of the form a" n-0 (a) Find the recurrence relations for the coefficients of the power series. 3 marks] (b) Determine the radius of convergence of the power series. l mar (c) Write the first eight terms of the series solution with the coefficients written in terms of ao and ai 2 marks]
3. Consider the following differential equation 0o and a series solution to...
Q1 (10 points) Consider the differential equation ty" _ y = 0. a) is this differential equation linear? What is its order? Is it homogeneous? b) Try a solution of the form y=x". Is this a solution for some r? If so, find all such r. c) Based on your answer to a) about linearity and b) about what y=x" are solutions, make an educated guess a the general solution looks like. Try that guess and check that it works....
Differential equation
1. Chapter 4 covers differential equations of the form an(x)y("4a-,(x)ye-i) + +4(x)y'+4(x)-g(x) Subject to initial conditions y)oyy-Co) Consider the second order differential equation 2x2y" + 5xy, + y-r-x 2- The Existence of a Unique Solution Theorem says there will be a unique solution y(x) to the initial-value problem at x=而over any interval 1 for which the coefficient functions, ai (x) (0 S is n) and g(x) are continuous and a, (x)0. Are there any values of x for...
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