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(i) We consider a one-dimensional potential barrier problem. In order for the particle to tunnel through the potential barrier of the width L, the difference between the barrier height U and the incident energy E of the particle with mass m has to be close. Using the transmission probability given in the text book / lecture, obtain the energy difference U-E which gives the transmission probability of exp(-2). (ii) We consider an infinite square well potential with the width L. Obtain the energy E_{gr} of the lowest energy level (ground state) of the particle with mass m, and show that E_{gr} scales linearly with E-U in the problem (i). The potential structures of (i) and (ii) can be viewed as "shadows" of each other. Energy U E- Electron 0 L (iii) We now consider a 3-dimensional infinite square well potential having the length of the x, y, and z directions to be all L. V= L**3 is the volume of the cube of this potential. We consider energy level of a single particle (boson) confined in this cube. Write down energy and degeneracy of the 8 quantum levels from the lowest energy level identified by the three quantum numbers to be (1,1,1). Make a table of quantum numbers, energy and degeneracy.

(i) We consider a one-dimensional potential barrier problem. In order for the particle to tunnel through the potential barrier of the width L, the difference between the barrier height U and the incident energy E of the particle with mass m has to be close. Using the transmission probability given in the text book / lecture, obtain the energy difference U-E which gives the transmission probability of exp(-2). (ii) We consider an infinite square well potential with the width L. Obtain the energy E_{gr} of the lowest energy level (ground state) of the particle with mass m, and show that E_{gr} scales linearly with E-U in the problem (i). The potential structures of (i) and (ii) can be viewed as "shadows" of each other. Energy U E- Electron 0 L (iii) We now consider a 3-dimensional infinite square well potential having the length of the x, y, and z directions to be all L. V= L**3 is the volume of the cube of this potential. We consider energy level of a single particle (boson) confined in this cube. Write down energy and degeneracy of the 8 quantum levels from the lowest energy level identified by the three quantum numbers to be (1,1,1). Make a table of quantum numbers, energy and degeneracy.

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(iv) Now we consider making V to be very large and putting many particles (total N particles) in this volume V. We now consider the case for the particles to be electrons, and consider the effect of spin, and Pauli principle. We consider the ideal case where there is no interaction between electrons, so they can be treated as a gas of free particles. We first consider that the energy of the particle can be given as (1/2)p**2/m, i.e., the energy of non-relativistic free particle. Obtain the maximum energy E_{max} of an electron at temperature T = 0 for the volume V and number of particles N. Also, Obtain the ratio E_{av}/E_{max} of the average kinetic energy E_{av} versus E_{max}. (v) We now consider the case when the kinetic energy of electrons is proportional to the momentump to the power of 1.5. Obtain the ratio E_{av}/E_{max} of the average kinetic energy E_{av} of this new gas with respect to the electron having the highest kinetic energy E_{max} at T = 0. (vi) In (iv) and (v), how the ratio E_{av}/E_ {max} would change if we consider 2-dimensional (2-d) electron gases, instead of 3-d ? Give two answers for the case (iv) with the kinetic energy proportional to p**2 and for (v) with the kinetic energy proportional to p**(1.5).
Transcribed Image Text:(iv) Now we consider making V to be very large and putting many particles (total N particles) in this volume V. We now consider the case for the particles to be electrons, and consider the effect of spin, and Pauli principle. We consider the ideal case where there is no interaction between electrons, so they can be treated as a gas of free particles. We first consider that the energy of the particle can be given as (1/2)p**2/m, i.e., the energy of non-relativistic free particle. Obtain the maximum energy E_{max} of an electron at temperature T = 0 for the volume V and number of particles N. Also, Obtain the ratio E_{av}/E_{max} of the average kinetic energy E_{av} versus E_{max}. (v) We now consider the case when the kinetic energy of electrons is proportional to the momentump to the power of 1.5. Obtain the ratio E_{av}/E_{max} of the average kinetic energy E_{av} of this new gas with respect to the electron having the highest kinetic energy E_{max} at T = 0. (vi) In (iv) and (v), how the ratio E_{av}/E_ {max} would change if we consider 2-dimensional (2-d) electron gases, instead of 3-d ? Give two answers for the case (iv) with the kinetic energy proportional to p**2 and for (v) with the kinetic energy proportional to p**(1.5).
(i) We consider a one-dimensional potential barrier problem. In order for the particle to tunnel through the potential barrier of the width L, the difference between the barrier height U and the incident energy E of the particle with mass m has to be close. Using the transmission probability given in the text book / lecture, obtain the energy difference U-E which gives the transmission probability of exp(-2). (ii) We consider an infinite square well potential with the width L. Obtain the energy E_{gr} of the lowest energy level (ground state) of the particle with mass m, and show that E_{gr} scales linearly with E-U in the problem (i). The potential structures of (i) and (ii) can be viewed as "shadows" of each other. Energy U ---E« Electron X L L (iii) We now consider a 3-dimensional infinite square well potential having the length of the x, y, and z directions to be all L. V=L**3 is the volume of the cube of this potential. We consider energy level of a single particle (boson) confined in this cube. Write down energy and degeneracy of the 8 quantum levels from the lowest energy level identified by the three quantum numbers to be (1,1,1). Make a table of quantum numbers, energy and degeneracy.
Transcribed Image Text:(i) We consider a one-dimensional potential barrier problem. In order for the particle to tunnel through the potential barrier of the width L, the difference between the barrier height U and the incident energy E of the particle with mass m has to be close. Using the transmission probability given in the text book / lecture, obtain the energy difference U-E which gives the transmission probability of exp(-2). (ii) We consider an infinite square well potential with the width L. Obtain the energy E_{gr} of the lowest energy level (ground state) of the particle with mass m, and show that E_{gr} scales linearly with E-U in the problem (i). The potential structures of (i) and (ii) can be viewed as "shadows" of each other. Energy U ---E« Electron X L L (iii) We now consider a 3-dimensional infinite square well potential having the length of the x, y, and z directions to be all L. V=L**3 is the volume of the cube of this potential. We consider energy level of a single particle (boson) confined in this cube. Write down energy and degeneracy of the 8 quantum levels from the lowest energy level identified by the three quantum numbers to be (1,1,1). Make a table of quantum numbers, energy and degeneracy.
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