Concept explainers
The expression for the allowed energies of particle and the ground state energy for an electron.
Answer to Problem 8P
The expression for the allowed energies of particle is
Explanation of Solution
Write the expression for momentum given by de Broglie relation.
Here,
Here,
Write the expression for energy given by de Broglie relation.
Here,
Assume that particle is a wave and it must vanish at boundaries of the box so the expression for wavelength is:
Write the expression for length of particle box in terms of wavelength.
From equation (III) and equation (IV), it can be concluded that:
Similarly,
Take the square of momentum in equation (I).
Substitute
Substitute
Substitute
Conclusion:
When the electron is in the lowest energy state is
Substitute
Thus, the expression for the allowed energies of particle is
Want to see more full solutions like this?
Chapter 8 Solutions
Modern Physics
- What is the ground state energy (in eV) of a proton confined to a one-dimensional box the size of the uranium nucleus that has a radius of approximately 15.0 fm?arrow_forwardConsider a very simplistic model of atomic nucleus in 1D: a proton is completely localized in a 1D box of width L = 1.00 × 10¬14m. In other words, the proton wavefunction outside of the "nucleus" is zero. Note that L represents a typical nuclear radius. (A) What are the energies of the ground and the first excited states? If the proton makes a transition from the first excited state to the ground state, what is the angular frequency of the emitted photon? (B) What is the probability that the proton in its ground state (i.e., the lowest energy state) is not found in the distance L/12 around each boundary of the box? (C) Using the uncertainty principle, derive a minimum possible value on the momentum uncertainty in the second state above the ground state. (D) Compare your answer to the previous question (B) to probability distribution one would obtain for a classical particle. First argue about how the probability distribution would look for a classical object in its ground state. How…arrow_forwardA certain atom requires 3.0 eV of energy to excite an electron from the ground level to the first excited level. Model the atom as an electron in a box and find the width L of the box.arrow_forward
- The general solution of the Schrodinger equation for a particle confined in an infinite square-well potential (where V = 0) of width L is w(x)= C sin kx + Dcos kx V2mE k where C and D are constants, E is the energy of the particle and m is the mass of the particle. Show that the energy E of the particle inside the square-well potential is quantised.arrow_forwardA particle of mass 1.60 x 10-28 kg is confined to a one-dimensional box of length 1.90 x 10-10 m. For n = 1, answer the following. (a) What is the wavelength (in m) of the wave function for the particle? m (b) What is its ground-state energy (in eV)? eV (c) What If? Suppose there is a second box. What would be the length L (in m) for this box if the energy for a particle in the n = 5 state of this box is the same as the ground-state energy found for the first box in part (b)? m (d) What would be the wavelength (in m) of the wave function for the particle in that case? marrow_forwardAn atom with 2 neutrons, 1 proton, 1 electron, is in its ground state when one of its neutrons undergoes a nuclear decay β (n → p + e + ν). The produced electron is fired at high speed, while the proton remains confined in the nucleus, forming a Helium nucleus with the original electronspinning around him. Find the probability that the ion resulting from He + is in the state 1S.arrow_forward
- Consider a model thermodynamic assembly in which the allowed one-particle states have energies 0, ?, 2?, 3?, 4?,5?,6?,.... The assembly has three particles and a total energy of 7?. Identify the possible particle number distributions and calculate the average distribution of the three particles in the energy states when the particles are (a) localized and distinguishable (b) gaseous bosons (c) gaseous fermionsarrow_forwardA particle with mass m is moving in three-dimensions under the potential energy U(r), where r is the radial distance from the origin. The state of the particle is given by the time-independent wavefunction, Y(r) = Ce-kr. Because it is in three dimensions, it is the solution of the following time-independent Schrodinger equation dıp r2 + U(r)µ(r). dr h2 d EÞ(r) = 2mr2 dr In addition, 00 1 = | 4ar?y? (r)dr, (A(r)) = | 4r²p²(r)A(r)dr. a. Using the fact that the particle has to be somewhere in space, determine C. Express your answer in terms of k. b. Remembering that E is a constant, and the fact that p(r) must satisfy the time-independent wave equation, what is the energy E of the particle and the potential energy U(r). (As usual, E and U(r) will be determined up to a constant.) Express your answer in terms of m, k, and ħ.arrow_forwardFor a classical system of non-interacting particles in the presence of a spherically symmetric potential V (r ) = yr, what is the mean energy per particle? y is a constant.arrow_forward
- Assume that an atomic nucleus can be thought of as a three-dimensional box with a width of 2 x 10^-14 m. If a proton moving as particles in this box, specify : Energy is excited first and second.arrow_forwardA three-dimensional wave function of a particle is u(x)=c/r exp(-kr/i)calculate the probability currentdensityarrow_forwardA 3.0 MeV proton is incident on a potential energy barrier of thickness 10 fm and height 10 MeV.What are (a) the transmission coefficient T, (b) the kinetic energy Kt the proton will have on the other side of the barrier if it tunnels through the barrier, and (c) the kinetic energy Kr it will have if it reflects from the barrier? A 3.0 MeV deuteron (the same charge but twice the mass as a proton) is incident on the same barrier.What are (d) T, (e) Kt, and (f) Kr?arrow_forward
- Principles of Physics: A Calculus-Based TextPhysicsISBN:9781133104261Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningPhysics for Scientists and Engineers: Foundations...PhysicsISBN:9781133939146Author:Katz, Debora M.Publisher:Cengage LearningModern PhysicsPhysicsISBN:9781111794378Author:Raymond A. Serway, Clement J. Moses, Curt A. MoyerPublisher:Cengage Learning
- Classical Dynamics of Particles and SystemsPhysicsISBN:9780534408961Author:Stephen T. Thornton, Jerry B. MarionPublisher:Cengage LearningUniversity Physics Volume 3PhysicsISBN:9781938168185Author:William Moebs, Jeff SannyPublisher:OpenStax