Consider an electron in a cubic box that measures 1nm on an edge a) Calculate the...
Consider an electron in a cubic box that measures 1nm on an
edge
a) Calculate the energy difference between the ground and first
excited states and compare this energy difference with
KbT at 300 K.
b) Using the Boltzman factor, Nx=N0 exp
(-delta E/KbT), calculate and comment on the relative
population of the first excited state at this temperature.
c) What minimum wavelength is required to excite the electron into
the the first excited state
d) How would you answer in part a) change if the symmetry of the
box was broken by setting the dimensions equal to x = 1 nm, y = 2
nm, and z = 3 nm? Show explicitly which energy state is the first
excited state.
Homework Answers
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Consider an electron in a cubic box that measures 1nm on an
edge
a) Calculate the...
An electron is confined in the ground state in a one-dimensional box of width 10-10 m....
An electron is confined in the ground state in a one-dimensional box of width 10-10 m. Its energy is known to be 38 eV. (a) Calculate the energy of the electron in its first and second excited states (b) Sketch the wave functions for the ground state, the first and the second excited states (c) Estimate the average force (in Newtons) exerted on the walls of the box when the electron is in the ground state. (d) Sketch the new...
2. (a) When a particle of mass 1.0 x 10-26 g in a one-dimensional box goes...
2. (a) When a particle of mass 1.0 x 10-26 g in a one-dimensional box goes from the n=3 level to n=1 level, it emits a radiation with frequency 5.0 x 1014 Hz. Calculate the length of the box. (b) Suppose that an electron freely moves around inside of a three-dimensional rectangular box with dimensions of 0.4 nm (width), 0.4 nm (length), and 0.5 nm (height). Calculate the frequency of the radiation that the electron would absorb during its transition...
Consider an electron in a one-dimensional box as a model of a quantum dot. Suppose the...
Consider an electron in a one-dimensional box as a model of a quantum dot. Suppose the box has width 0.7 nm. For this problem, absorption of light and subsequent relaxation connect two states (i andj) with a difference in energy, AEi E - E. (a) Calculate AEsi and AE2I for luminescence from excited energy levels to the ground state. Convert the energies to the corresponding wavelengths of light, λ31 and λ21. (b) Find the wavelength of light that corresponds to...
Consider an electron in a one-dimensional box of length 0.16 nm. (a) Calculate the energy difference...
Consider an electron in a one-dimensional box of length 0.16 nm. (a) Calculate the energy difference between the n = 2 and n = 1 states of the electron. (b) Calculate the energy difference for a N2 molecule in a one-dimensional box of length 11.2 cm.
An electron is confined to a box of width 10 nm. How much energy must be...
An electron is confined to a box of width 10 nm. How much energy must be acquired to boost it into the first excited state, n=2, from the ground state?
Be3+ is a one-electron ion. It is a beryllium atom that has lost 3 of its...
Be3+ is a one-electron ion. It is a beryllium atom that has lost 3 of its electrons. Minimum energy of a photon that would ionize this ion= -217.6 eV The wavelength of the photon that would excite Be3+ from its ground state to its first excited state= 7.56 nm Calculate the kinetic energy of the photoelectrons if radiation of 4.0 nm of wavelength falls on an ensemble of Be3+ ions.
| The principle quantum number n characterizes the electronic interactions between the electron and the nucleus...
| The principle quantum number n characterizes the electronic interactions between the electron and the nucleus of an H-atom. (5 points) 1. Calculate the longest wavelength absorption (in nm) of the electron at the ground state. 2. Calculate the energy difference between the ground state and the 1st excited state. 3. Calculate the ratio of the probability of finding the electron at the first excited state over that at the ground state at 25°C. 4. Calculate the ratio of the...
2. Consider an electron in a 1D potential box (V(x) = 0 for 0<x<L, V(x) =...
2. Consider an electron in a 1D potential box (V(x) = 0 for 0<x<L, V(x) = co otherwise) of length L = 1 nm. The electron is described by the wave function, c) = Jasin ( (a) Using the appropriate Hamiltonian derive an expression for the kinetic energy of the electron (5 marks) (b) Calculate the energy (in Joules) of the transition between the ground state and the 1 excited state. [3 marks]
Calculate the probability of exciting an electron in a one-dimensional box (actually a nanoscale wire) to...
Calculate the probability of exciting an electron in a one-dimensional box (actually a nanoscale wire) to the n 2 excited state if the box is 10.0 nm long and the temperature is 410.0 K. For the one- dimensional box, En = n2t2h2/(2ma2) and the levels are non-degenerate (but remember that the energy should be measured relative to the ground state). For this example, T2h/(2mea2) is equal to 1.381- 10 23 JK-1 J and the partition function is 2.44. kB 6.02-10-22...
An electron (mass m) is trapped ina 2-dimensional infinite square box of sides Lx - L...
An electron (mass m) is trapped ina 2-dimensional infinite square box of sides Lx - L - L. Take Eo = 92/8mL2. Consider the first four energy levels: the ground state and the first three excited states. 1) Calculate the ground-state energy in terms of Ep. (That is, the ground-state energy is what multiple of Eo? Eo Submit 2) In terms of Eo, what is the energy of the first excited state? (That is, the energy of the first excited...
An electron is confined in the ground state in a one-dimensional box of width 10-10 m. Its energy is known to be 38 eV. (a) Calculate the energy of the electron in its first and second excited states (b) Sketch the wave functions for the ground state, the first and the second excited states (c) Estimate the average force (in Newtons) exerted on the walls of the box when the electron is in the ground state. (d) Sketch the new...
Consider an electron in a one-dimensional box as a model of a quantum dot. Suppose the box has width 0.7 nm. For this problem, absorption of light and subsequent relaxation connect two states (i andj) with a difference in energy, AEi E - E. (a) Calculate AEsi and AE2I for luminescence from excited energy levels to the ground state. Convert the energies to the corresponding wavelengths of light, λ31 and λ21. (b) Find the wavelength of light that corresponds to...
| The principle quantum number n characterizes the electronic interactions between the electron and the nucleus of an H-atom. (5 points) 1. Calculate the longest wavelength absorption (in nm) of the electron at the ground state. 2. Calculate the energy difference between the ground state and the 1st excited state. 3. Calculate the ratio of the probability of finding the electron at the first excited state over that at the ground state at 25°C. 4. Calculate the ratio of the...
2. Consider an electron in a 1D potential box (V(x) = 0 for 0<x<L, V(x) = co otherwise) of length L = 1 nm. The electron is described by the wave function, c) = Jasin ( (a) Using the appropriate Hamiltonian derive an expression for the kinetic energy of the electron (5 marks) (b) Calculate the energy (in Joules) of the transition between the ground state and the 1 excited state. [3 marks]
Calculate the probability of exciting an electron in a one-dimensional box (actually a nanoscale wire) to the n 2 excited state if the box is 10.0 nm long and the temperature is 410.0 K. For the one- dimensional box, En = n2t2h2/(2ma2) and the levels are non-degenerate (but remember that the energy should be measured relative to the ground state). For this example, T2h/(2mea2) is equal to 1.381- 10 23 JK-1 J and the partition function is 2.44. kB 6.02-10-22...
An electron (mass m) is trapped ina 2-dimensional infinite square box of sides Lx - L - L. Take Eo = 92/8mL2. Consider the first four energy levels: the ground state and the first three excited states. 1) Calculate the ground-state energy in terms of Ep. (That is, the ground-state energy is what multiple of Eo? Eo Submit 2) In terms of Eo, what is the energy of the first excited state? (That is, the energy of the first excited...
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