Graph the potential energy function (with respect to x), U(x), of your oscillator and use this...
Graph the potential energy function (with respect to x), U(x), of your oscillator and use this to give a physical explanation of your observation. U(x) can be directly obtained from the equation of motion (eq (9)). Show clear working for how you derive this.
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Graph the potential energy function (with respect to
x), U(x), of your oscillator and use this...
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...
A particle is introduced to a region with a potential described by U(x)--2x2 +x*+1 Joules. 3. a. ...
A particle is introduced to a region with a potential described by U(x)--2x2 +x*+1 Joules. 3. a. (2 pts) In software, plot the potential U) Set your axis ranges: -2 SxS2 and 0s b. (5 pts) Find the equilibrium positions and determine whether they are stable or c. (8 pts) Describe the motion of the particle for total energy values E-О.0.05. 1.0, 2.0 unstable. Explain how you arrived at your answers. (all in Joules). What I am looking for here...
A particle of mass m and energy E moving along the x axis is subjected to a potential energy function U(x).
\((25\) marks) A particle of mass \(m\) and energy \(E\) moving along the \(x\) axis is subjected to a potential energy function \(U(x) .\) (a) Suppose \(\psi_{1}(x)\) and \(\psi_{2}(\mathrm{x})\) are two wave functions of the system with the same energy \(E .\) Derive an expression to relate \(\psi_{1}(x), \psi_{2}(x)\), and their derivatives. (b) By requiring the wave functions to vanish at infinity, show that \(\psi_{1}(x)\) and \(\psi_{2}(x)\) can at most differ by a multiplicative constant. Hence, what conclusion can you...
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...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x,...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x, t) = (x)e-1st/ where and E, are the harmonic oscillator's stationary states and their corresponding energies. (a) Show that the expectation value of position is (hint: use the results of Problem 4): (v) = A cos (wt - ) where the real constants A and o are given by: 1 2 Ae-id-1 " Entichtin Interpret this result, comparing it with the motion of a...
U(x) Goules) x (meters) The graph above shows the potential energy upx) of a particle as...
U(x) Goules) x (meters) The graph above shows the potential energy upx) of a particle as a function of its position x a. Identify all points of equilibrium for this partidle. Suppose the particle has a constant total energy of 4.0 joules, as shown by the dashed line on the graph. b. energy of the particle at the following positions x2.0 m li.x=4.0 m c. Can the particle reach the position x-0.5 m?。Yes。No
A particle moves and has a potential energy that can be described by the equation U(x)...
A particle moves and has a potential energy that can be described by the equation U(x) = 4 sin(2 x) where U(x) is in J. The total energy of the particle is E_tot = 2 J. Make a well-labelled graph of U(x)vs. x from x = 0 to x = pi. Draw a line corresponding to E_tot on your diagram. Assume the particle is moving in the positive x direction. Where is the particle speeding up? Make sure you solve...
Classical Mechanics Let us consider the following kinetic (T) and potential (U) energies of a two-dimensional...
Classical Mechanics Let us consider the following kinetic (T) and potential (U) energies of a two-dimensional oscillator : ?(?,̇ ?̇)= ?/2 (?̇²+ ?̇²) ?(?,?)= ?/2 (?²+?² )+??? where x and y denote, respectively, the cartesian displacements of the oscillator; ?̇= ??/?? and ?̇= ??/?? the time derivatives of the displacements; m the mass of the oscillator; K the stiffness constant of the oscillator; A is the coupling constant. 1) Using the following coordinate transformations, ?= 1/√2 (?+?) ?= 1/√2 (?−?)...
5. (10 points) A simple function that looks like the potential well of a diatomic molecule...
5. (10 points) A simple function that looks like the potential well of a diatomic molecule is the Morse potential given by: U(x) = D. (1-e-Bx) (1) where, x is the displacement of the bond from its equilibrium position, and D. is the value of U(x) at large separations. D. is called the classical dissociation energy and is characterized by the depth of the potential well. We can expand U(x) in a Taylor series about x = 0 to obtain...
Identify which potential energy functions 𝑈 cannot be a potential energy function of a simple harmonic oscillator.
Identify which potential energy functions 𝑈 cannot be a potential energy function of a simple harmonic oscillator.
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...
A particle is introduced to a region with a potential described by U(x)--2x2 +x*+1 Joules. 3. a. (2 pts) In software, plot the potential U) Set your axis ranges: -2 SxS2 and 0s b. (5 pts) Find the equilibrium positions and determine whether they are stable or c. (8 pts) Describe the motion of the particle for total energy values E-О.0.05. 1.0, 2.0 unstable. Explain how you arrived at your answers. (all in Joules). What I am looking for here...
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...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x, t) = (x)e-1st/ where and E, are the harmonic oscillator's stationary states and their corresponding energies. (a) Show that the expectation value of position is (hint: use the results of Problem 4): (v) = A cos (wt - ) where the real constants A and o are given by: 1 2 Ae-id-1 " Entichtin Interpret this result, comparing it with the motion of a...
U(x) Goules) x (meters) The graph above shows the potential energy upx) of a particle as a function of its position x a. Identify all points of equilibrium for this partidle. Suppose the particle has a constant total energy of 4.0 joules, as shown by the dashed line on the graph. b. energy of the particle at the following positions x2.0 m li.x=4.0 m c. Can the particle reach the position x-0.5 m?。Yes。No
A particle moves and has a potential energy that can be described by the equation U(x) = 4 sin(2 x) where U(x) is in J. The total energy of the particle is E_tot = 2 J. Make a well-labelled graph of U(x)vs. x from x = 0 to x = pi. Draw a line corresponding to E_tot on your diagram. Assume the particle is moving in the positive x direction. Where is the particle speeding up? Make sure you solve...
5. (10 points) A simple function that looks like the potential well of a diatomic molecule is the Morse potential given by: U(x) = D. (1-e-Bx) (1) where, x is the displacement of the bond from its equilibrium position, and D. is the value of U(x) at large separations. D. is called the classical dissociation energy and is characterized by the depth of the potential well. We can expand U(x) in a Taylor series about x = 0 to obtain...
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