The one-dimensional infinite potential well can be generalized to three dimensions.
(25 marks) The one-dimensional infinite potential well can be generalized to three dimensions. The allowed energies for a particle of mass \(m\) in a cubic box of side \(L\) are given by
$$ E_{n_{p} n_{r, n_{i}}}=\frac{\pi^{2} \hbar^{2}}{2 m L^{2}}\left(n_{x}^{2}+n_{y}^{2}+n_{z}^{2}\right) \quad\left(n_{x}=1,2, \ldots ; n_{y}=1,2, \ldots ; n_{z}=1,2, \ldots\right) $$
(a) If we put four electrons inside the box, what is the ground-state energy of the system? Here the ground-state energy is defined to be the minimum energy of the system of electrons. You can neglect the electrostatic interactions among the electrons in this problem. (b) If we use this model to describe multi-electron atoms, what are the atomic numbers of the lightest three noble gases? (c) The atomic number of titanium is \(22 .\) What is the ground-state energy of titanium according to this model?
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The one-dimensional infinite potential well can be generalized to three dimensions.
16. Suppose that five electrons are placed in a one-dimensional infinite potential well of length L....
16. Suppose that five electrons are placed in a one-dimensional infinite potential well of length L. What is the energy of the ground state of electrons? What tb diore the Cour this system of five electrons? What is of the ground state? Take the exclusion principle into account, and ignore the Cou- lomb interaction of the electrons with each other.
The one-dimensional Schrindinger wave equation for a particle in a potential field
The one-dimensional Schrindinger wave equation for a particle in a potential field \(V=\) \(\frac{1}{2} k x^{2}\) is$$ -\frac{h^{2}}{2 m} \frac{d^{2} \psi}{d x^{2}}+\frac{1}{2} k x^{2} \psi=E \psi(x) $$(a) Lsing \(\xi=\alpha x\) and a constant \(\lambda\), we have$$ a=\left(\frac{m k}{A^{2}}\right)^{1 / 4}, \quad A=\frac{2 L}{A}\left(\frac{m}{k}\right)^{1 / 2} $$show that$$ \frac{d^{2} y(\xi)}{d \xi^{2}}+\left(\lambda-\xi^{2}\right) \psi(\xi)=0 $$(b) Substituting$$ \psi(\xi)=y(\xi) e^{2} / 2 $$show that \(y(t)\) satisfies the Hermite di fferential equation.
Consider two Einstein solids A and B that can exchange energy (but not oscillators/particles) wit...
Using matlab, evaluate the following system:Consider two Einstein solids \(A\) and \(B\) that can exchange energy (but not oscillators/particles) with one another but the combined composite system is isolated from the surroundings. Suppose systems \(A\) and \(B\) have \(N_{A}\) and \(N_{B}\) oscillators, and \(q_{A}\) and \(q_{B}\) units of energy respectively. The total number of microstates for this macrostate for the macrostate \(N_{A}, N_{B}, q, q_{A}\) is given by$$ \Omega\left(N_{A}, N_{B}, q, q_{A}\right)=\Omega\left(N_{A}, q_{A}\right) \Omega\left(N_{B}, q_{B}\right) $$where$$ \Omega\left(N_{i}, q_{i}\right)=\frac{\left(q_{i}+N_{i}-1\right) !}{q_{i} !\left(N_{i}-1\right)...
Seven identical particles are placed in a one-dimensional well with infinite potential: the spatial size of...
Seven identical particles are placed in a one-dimensional well with infinite potential: the spatial size of the well is L = 1 nm. Calculate the energy of the base state for the system, if the particles are a) electrons b) pions (which are bosons) with mass = 264 and electron mass
Suppose that an electron is trapped in a one- dimensional, infinite potential well of width 250...
Suppose that an electron is trapped in a one- dimensional, infinite potential well of width 250 nm is excited from the 2nd excited state to the fifth excited state. What energy must be transferred to the electron in order to make this transition? Answer: 1.62 x 10^-4 eV Check Correct Marks for this submission: 2.00/2.00. What wavelength photon does this correspond to? Answer: 75.15*10^-4m Check Considering all of the possible ways that the excited electron can de-excite back down to...
A NON stationary state A particle of mass m is in an infinite square well potential of width L, a...
A NON stationary state A particle of mass m is in an infinite square well potential of width L, as in McIntyre's section 5.4. Suppose we have an initial state vector lv(t -0) results from Mclntrye without re-deriving them, and you may use a computer for your math as long as you include your code in your solution A(3E1) 4iE2)). You may use E. (4 pts) Use a computer to plot this probability density at 4 times: t 0, t2...
(15 points) Encounter with a semi-infinite potential "well" In this problem we will investigate one situation involving a a semi-infinite one-dimensional po- tential well (Figure 1) U=0 regio...
(15 points) Encounter with a semi-infinite potential "well" In this problem we will investigate one situation involving a a semi-infinite one-dimensional po- tential well (Figure 1) U=0 region 1 region 2 region 3 Figure 1: Semi-infinite potential for Problem 3 This potential is piecewise defined as follows where Uo is some positive value of energy. The three intervals in x have been labeled region 1,2 and 3 in Figure 1 Consider a particle of mass m f 0 moving in...
The ideal gas law, discovered experimentally, is an equation of state that relates the observable state variables of the gas. pressure, temperature, and density
The ideal gas law, discovered experimentally, is an equation of state that relates the observable state variables of the gas. pressure, temperature, and density (or quantity per volume$$ \eta V=N k_{\mathrm{B}} T(\mathrm{or} p V=n \mathrm{RT}) $$Where \(N\) is the number of atoms, \(n\) is the number of moles, and \(R\) and \(k_{\mathrm{B}}\) are ideal gas constants such that \(R=N_{\mathrm{A}} k_{\mathrm{B}}\), where \(N_{A}\) is Avogadro's number. In this problem. you should use Boltzmann's constant instead of the gas constant \(R\).Remaıkably. the...
The eigenfunctions for a particle in a one-dimensional box of length L, and the corresponding energy...
The eigenfunctions for a particle in a one-dimensional box of length L, and the corresponding energy eigenvalues are given below. What is the variance of measurements for the linear momentum, i.e., Op = v<p? > - <p>2? Øn (x) = ( )" sin nga, n= 1, 2,.. En = n2h2 8m12 Note the Hamiltonian operator to give the energy is H = = - 42 8n72 dx2 nh 2L oo O nềh2 412 Uncertain since x is known. Following Question...
2. Goal of this problem is to study how tunnelling in a two-well system emerges. In particular, w...
2. Goal of this problem is to study how tunnelling in a two-well system emerges. In particular, we are interested in determining how the tunnelling rate T' of a particle with mass m scales as a function of the (effective) height Vo - E and width b of an energy barrier separating the two wells. The following graphics illustrates the set-up. Initially the particle may be trapped on the left side corresponding to the state |L〉, we are now interested...
16. Suppose that five electrons are placed in a one-dimensional infinite potential well of length L. What is the energy of the ground state of electrons? What tb diore the Cour this system of five electrons? What is of the ground state? Take the exclusion principle into account, and ignore the Cou- lomb interaction of the electrons with each other.
Using matlab, evaluate the following system:Consider two Einstein solids \(A\) and \(B\) that can exchange energy (but not oscillators/particles) with one another but the combined composite system is isolated from the surroundings. Suppose systems \(A\) and \(B\) have \(N_{A}\) and \(N_{B}\) oscillators, and \(q_{A}\) and \(q_{B}\) units of energy respectively. The total number of microstates for this macrostate for the macrostate \(N_{A}, N_{B}, q, q_{A}\) is given by$$ \Omega\left(N_{A}, N_{B}, q, q_{A}\right)=\Omega\left(N_{A}, q_{A}\right) \Omega\left(N_{B}, q_{B}\right) $$where$$ \Omega\left(N_{i}, q_{i}\right)=\frac{\left(q_{i}+N_{i}-1\right) !}{q_{i} !\left(N_{i}-1\right)...
Suppose that an electron is trapped in a one- dimensional, infinite potential well of width 250 nm is excited from the 2nd excited state to the fifth excited state. What energy must be transferred to the electron in order to make this transition? Answer: 1.62 x 10^-4 eV Check Correct Marks for this submission: 2.00/2.00. What wavelength photon does this correspond to? Answer: 75.15*10^-4m Check Considering all of the possible ways that the excited electron can de-excite back down to...
A NON stationary state A particle of mass m is in an infinite square well potential of width L, as in McIntyre's section 5.4. Suppose we have an initial state vector lv(t -0) results from Mclntrye without re-deriving them, and you may use a computer for your math as long as you include your code in your solution A(3E1) 4iE2)). You may use E. (4 pts) Use a computer to plot this probability density at 4 times: t 0, t2...
(15 points) Encounter with a semi-infinite potential "well" In this problem we will investigate one situation involving a a semi-infinite one-dimensional po- tential well (Figure 1) U=0 region 1 region 2 region 3 Figure 1: Semi-infinite potential for Problem 3 This potential is piecewise defined as follows where Uo is some positive value of energy. The three intervals in x have been labeled region 1,2 and 3 in Figure 1 Consider a particle of mass m f 0 moving in...
The ideal gas law, discovered experimentally, is an equation of state that relates the observable state variables of the gas. pressure, temperature, and density (or quantity per volume$$ \eta V=N k_{\mathrm{B}} T(\mathrm{or} p V=n \mathrm{RT}) $$Where \(N\) is the number of atoms, \(n\) is the number of moles, and \(R\) and \(k_{\mathrm{B}}\) are ideal gas constants such that \(R=N_{\mathrm{A}} k_{\mathrm{B}}\), where \(N_{A}\) is Avogadro's number. In this problem. you should use Boltzmann's constant instead of the gas constant \(R\).Remaıkably. the...
The eigenfunctions for a particle in a one-dimensional box of length L, and the corresponding energy eigenvalues are given below. What is the variance of measurements for the linear momentum, i.e., Op = v<p? > - <p>2? Øn (x) = ( )" sin nga, n= 1, 2,.. En = n2h2 8m12 Note the Hamiltonian operator to give the energy is H = = - 42 8n72 dx2 nh 2L oo O nềh2 412 Uncertain since x is known. Following Question...
2. Goal of this problem is to study how tunnelling in a two-well system emerges. In particular, we are interested in determining how the tunnelling rate T' of a particle with mass m scales as a function of the (effective) height Vo - E and width b of an energy barrier separating the two wells. The following graphics illustrates the set-up. Initially the particle may be trapped on the left side corresponding to the state |L〉, we are now interested...
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