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6. A proton of mass m inside a nucleus is in a potential U(r) = A...
Rutherford's scattering experiments gave the first indications that an atom consists of a small, dense, positively charg...
Rutherford's scattering experiments gave the first indications that an atom consists of a small, dense, positively charged nucleus surrounded by negatively charged electrons. His experiments also allowed for a rough determination of the size of the nucleus. In this problem, you will use the uncertainty principle to get a rough idea of the kinetic energy of a particle inside the nucleus. Consider a nucleus with a diameter of roughly 5.0×10−15 meters. The uncertainty Δp sets a lower bound on the...
The potential energy of two atoms in a diatomic molecule is approximated by U(r)=a/r^12−b/r^6, where r...
The potential energy of two atoms in a diatomic molecule is approximated by U(r)=a/r^12−b/r^6, where r is the spacing between atoms and a and b are positive constants. Find the component of force along the line connecting the two atoms, Fr(r), on one atom as a function of r. Find the equilibrium distance between the two atoms. Is this equilibrium stable? Suppose the distance between the two atoms is equal to the equilibrium distance found in part A. What minimum...
The potential energy of two atoms in a diatomic molecule is approximated by U(r)=d/r12-b/r6, where r...
The potential energy of two atoms in a diatomic molecule is approximated by U(r)=d/r12-b/r6, where r is the spacing between atoms and a and b are positive constants. Find the force F(r) on one atom as a function of r. Find the equilibrium distance between the two atoms. Express your answer in terms of the variables a and b. Is this equilibrium stable? Suppose the distance between the two atoms is equal to the equilibrium distance found in part (b)....
A particle of mass m moves in one dimension. Its potential energy is given by U(x)...
A particle of mass m moves in one dimension. Its potential energy is given by U(x) = -Voe-22/22 where U, and a are constants. (a) Draw an energy diagram showing the potential energy U(). Choose some value for the total mechanical energy E such that -U, < E < 0. Mark the kinetic energy, the potential energy and the total energy for the particle at some point of your choosing. (b) Find the force on the particle as a function...
2. The interaction between neutral atoms can be approximated with the Lennard- Jones potential U(r)-4e where...
2. The interaction between neutral atoms can be approximated with the Lennard- Jones potential U(r)-4e where r is the interatomic distance, and E and σ are constants. a) If U is measured in Joules and r is measured in nanometers (nm), what are the units of є and σ. b) What is the interatomic distance at which the potential is zero?. c) Find the value of r for which U is a minimum d) What is the minimum value of...
Recall that an energy eigenfunction of any central potential V (r) may be writtren as ψn`m(r,...
Recall that an energy eigenfunction of any central potential V
(r) may be writtren as ψn`m(r, θ, φ) = Rn`(r)Y`m(θ, φ). This
problem explores the behavior of ψ in the vicinity of the origin r
= 0. Recall that the function u(r) = rRn`(r) satisfies the
equation
− ~ 2 2m d 2u dr2 + ~ 2 `(` + 1) 2mr2 + V (r) u = Eu, (1)
where E is the energy eigenvalue. Note that Eq. (1) has the...
The interaction potential between 2 particles that generates the desired effect of particles coming together, sticking,...
The interaction potential between 2 particles that generates the desired effect of particles coming together, sticking, and bouncing off if knocked with sufficient energy is given by the Lennard-Jones (LJ) interaction potential. This potential as a function of distance r between any 2 particles is: U(n) = 4e (9)"-09)) (1) Consider a solution of gold nanoparticles of diameter o = 2 nm. Take e 1 eV. Note that 1 eV = 1.6 x 10-19 Joules. 2. At what distance between...
Chapter 28, Problem 079 A proton, a deuteron (q = +e, m = 2.0 u), and...
Chapter 28, Problem 079 A proton, a deuteron (q = +e, m = 2.0 u), and an alpha particle (q 2e, m 4.0 u) are accelerated through the same potential difference and then enter R the same region of uniform magnetic field ,moving perpendicular to . (a) What is the ratio of the proton's kinetic energy Ko to the alpha particle's kinetic energy K? (b) What is the ratio of deuteron's kinetic energy Kd to Ka? Assume that the radius...
1. The quantum states of a particle moving freely in a circle of radius r are...
1. The quantum states of a particle moving freely in a circle of radius r are described by (0) = Cewe where C is a constant, e denotes angle, n = 0, +1, +2,... is an integer identifying the quantum state of the particle, and wn is constant for a given n. a) Show that Un0 satisfies don d02 b) Find wn such that Un (@+ 2) = Un) c) Find the value of such that any two yn (0)...
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...
The potential energy of two atoms in a diatomic molecule is approximated by U(r)=d/r12-b/r6, where r is the spacing between atoms and a and b are positive constants. Find the force F(r) on one atom as a function of r. Find the equilibrium distance between the two atoms. Express your answer in terms of the variables a and b. Is this equilibrium stable? Suppose the distance between the two atoms is equal to the equilibrium distance found in part (b)....
A particle of mass m moves in one dimension. Its potential energy is given by U(x) = -Voe-22/22 where U, and a are constants. (a) Draw an energy diagram showing the potential energy U(). Choose some value for the total mechanical energy E such that -U, < E < 0. Mark the kinetic energy, the potential energy and the total energy for the particle at some point of your choosing. (b) Find the force on the particle as a function...
2. The interaction between neutral atoms can be approximated with the Lennard- Jones potential U(r)-4e where r is the interatomic distance, and E and σ are constants. a) If U is measured in Joules and r is measured in nanometers (nm), what are the units of є and σ. b) What is the interatomic distance at which the potential is zero?. c) Find the value of r for which U is a minimum d) What is the minimum value of...
Recall that an energy eigenfunction of any central potential V
(r) may be writtren as ψn`m(r, θ, φ) = Rn`(r)Y`m(θ, φ). This
problem explores the behavior of ψ in the vicinity of the origin r
= 0. Recall that the function u(r) = rRn`(r) satisfies the
equation
− ~ 2 2m d 2u dr2 + ~ 2 `(` + 1) 2mr2 + V (r) u = Eu, (1)
where E is the energy eigenvalue. Note that Eq. (1) has the...
The interaction potential between 2 particles that generates the desired effect of particles coming together, sticking, and bouncing off if knocked with sufficient energy is given by the Lennard-Jones (LJ) interaction potential. This potential as a function of distance r between any 2 particles is: U(n) = 4e (9)"-09)) (1) Consider a solution of gold nanoparticles of diameter o = 2 nm. Take e 1 eV. Note that 1 eV = 1.6 x 10-19 Joules. 2. At what distance between...
Chapter 28, Problem 079 A proton, a deuteron (q = +e, m = 2.0 u), and an alpha particle (q 2e, m 4.0 u) are accelerated through the same potential difference and then enter R the same region of uniform magnetic field ,moving perpendicular to . (a) What is the ratio of the proton's kinetic energy Ko to the alpha particle's kinetic energy K? (b) What is the ratio of deuteron's kinetic energy Kd to Ka? Assume that the radius...
1. The quantum states of a particle moving freely in a circle of radius r are described by (0) = Cewe where C is a constant, e denotes angle, n = 0, +1, +2,... is an integer identifying the quantum state of the particle, and wn is constant for a given n. a) Show that Un0 satisfies don d02 b) Find wn such that Un (@+ 2) = Un) c) Find the value of such that any two yn (0)...
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...
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