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1. (QM) For a particle of mass M confined to the region << with no other forces acting on it, solutions of the time independent Schrödinger equation (T.I.S.E.) of the following form exist. (x) = B sin(mxx/b) Use the boundary conditions for this problem to find the values of m for which this this is a valid solution of the time-independent Schrödinger equation (T.I.S.E.). Show by direct substitution that this is a solution of the T.I.S.E. and, hence, derive an expression for the energy of the particle in terms of b, and m and M. Normalise this wave function. For the lowest allowed value of m, estimate the probability of observing the particle within 10% of the middle the region. The transition m+2+m for an electron trapped in a 1-dimensional potential 2nm across produces light with a wavelength of 823 nm. What is the value of m for this transition?

1. (QM) For a particle of mass M confined to the region << with no other forces acting on it, solutions of the time independent Schrödinger equation (T.I.S.E.) of the following form exist. (x) = B sin(mxx/b) Use the boundary conditions for this problem to find the values of m for which this this is a valid solution of the time-independent Schrödinger equation (T.I.S.E.). Show by direct substitution that this is a solution of the T.I.S.E. and, hence, derive an expression for the energy of the particle in terms of b, and m and M. Normalise this wave function. For the lowest allowed value of m, estimate the probability of observing the particle within 10% of the middle the region. The transition m+2+m for an electron trapped in a 1-dimensional potential 2nm across produces light with a wavelength of 823 nm. What is the value of m for this transition?

University Physics Volume 3
University Physics Volume 3
17th Edition
ISBN:9781938168185
Author:William Moebs, Jeff Sanny
Publisher:William Moebs, Jeff Sanny
Chapter7: Quantum Mechanics
Section: Chapter Questions
Problem 12CQ: Explain the difference between time-dependent and independent SchrÖdinger's equations.
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1. (QM) For a particle of mass M confined to the region << with no other forces acting on it, solutions of the time independent Schrödinger equation (T.I.S.E.) of the following form exist. (x) = B sin(mxx/b) Use the boundary conditions for this problem to find the values of m for which this this is a valid solution of the time-independent Schrödinger equation (T.I.S.E.). Show by direct substitution that this is a solution of the T.I.S.E. and, hence, derive an expression for the energy of the particle in terms of b, and m and M. Normalise this wave function. For the lowest allowed value of m, estimate the probability of observing the particle within 10% of the middle the region. The transition m+2+m for an electron trapped in a 1-dimensional potential 2nm across produces light with a wavelength of 823 nm. What is the value of m for this transition?
Transcribed Image Text:1. (QM) For a particle of mass M confined to the region << with no other forces acting on it, solutions of the time independent Schrödinger equation (T.I.S.E.) of the following form exist. (x) = B sin(mxx/b) Use the boundary conditions for this problem to find the values of m for which this this is a valid solution of the time-independent Schrödinger equation (T.I.S.E.). Show by direct substitution that this is a solution of the T.I.S.E. and, hence, derive an expression for the energy of the particle in terms of b, and m and M. Normalise this wave function. For the lowest allowed value of m, estimate the probability of observing the particle within 10% of the middle the region. The transition m+2+m for an electron trapped in a 1-dimensional potential 2nm across produces light with a wavelength of 823 nm. What is the value of m for this transition?
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