What differences do you see from the one dimensional infinite
well?
Separation of Variables Try a solution where >(x,y) - P(x)Q(y). Substitute and show that 2 2Qdy P(x) dx2 h2 Argue that both sides must equal a constant and that constant should be negative if E >0.
The two dimensional infinite well In two dimensions the Schrodinger equation is 0 I지 늘.lyl 들 where V (x, y) =く00
Homework Answers
In solving the particle in a one dimensional infinite depth box problem (0k x < a)...
In solving the particle in a one dimensional infinite depth box problem (0k x < a) we started with the function following is a true statement? (a) The value of k is found by requiring that the solution be normalized. (b) The function wx) is not an eigenfunciton of the operator d2/dx2 (c) It is necessary that this function equals a when x=0 (ie, Ψ(0) = a). (d) The boundary condition at x = 0 is used to show that...
3. For a particle moving in an infinite, one-dimensional, symmetric square well of width 2a, show...
3. For a particle moving in an infinite, one-dimensional, symmetric square well of width 2a, show that the (normalized) wave functions are of the form ?-kx).va. cos?x): "-1. 3.5 ,.. COS ? -?? r")(x)=?sin n-r | ; n-2, 4, 6 Express the state ?(x)=N sin,(rx/a) as a linear superposition eigenstates, and find its normalization constant N. of the above HINT sin39-3sin ?-4sin'?
3 At a given time, the normalised wave function for a particle in a one-dimensional infinite...
3 At a given time, the normalised wave function for a particle in a one-dimensional infinite square well -a < x < a is given by 2 sin2 V inside the well and zero outside. Find the probability that a measurement of energy yields the eigenvalue En. (Hint: use data on page 6.) [6] Useful Data and Formulas = 1.60 x 10-19 C Elementary charge e h/2T=1.05 x 10-34 Js Planck's constant 3.00 x 108 m s-1 Speed of light...
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...
An electron in a one-dimensional infinite potential well of width L is found to have the...
An electron in a one-dimensional infinite potential well of width L is found to have the normalized wave function ψ(x)- sin(2 r ). (a) What is the probability of finding the electron within the interval from x=010 x = L/2 ? (b) At what position or positions is the electron most likely to be found? In other words, find the value(s) of x where the probability of finding the particle is the greatest?
onfined to move say, a typical size of a desk). Based on the nics, what is...
onfined to move say, a typical size of a desk). Based on the nics, what is the minimum speed of the student? quantum (c). What should the quantum number n be if the student is moving with a speed of 1.25 m/s? (d)What is the separation of the energy levels of the student moving with this speed (extra points: 10 points)The time independent Schrodinger equation in the infinite 5. potential well is d2) 2m Eψ 0 . + dx2 h2...
Q4. Consider the 1D infinite square-well potential shown in the figure below. V(x) O0 Position (a)...
Q4. Consider the 1D infinite square-well potential shown in the figure below. V(x) O0 Position (a) State the time-independent Schrödinger equation within the region 0<x<L for a particle with positive energy E 2 marks] (b) The wavefunction for 0<x< L can be written in the general form y(x) = Asin kx + B cos kx. Show that the normalised wavefunction for the 1D infinite potential well becomes 2sn'n? ?snT/where ( "1,2,3 ! where ( n = 1,2,5, ). [4 marks]...
Consider a particle confined in two-dimensional box with infinite walls at x 0, L;y 0, L. the dou...
quantum mechanics
Consider a particle confined in two-dimensional box with infinite walls at x 0, L;y 0, L. the doubly degenerate eigenstates are: Ιψη, p (x,y))-2sinnLx sinpry for 0 < x, y < L elsewhere and their eigenenergies are: n + p, n, p where n, p-1,2, 3,.... Calculate the energy of the first excited state up to the first order in perturbation theory due to the addition of: 2 2
Consider a particle confined in two-dimensional box with infinite...
(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...
Consider an 2D infinite square well that extends from 0 to a in the x direction,...
Consider an 2D infinite square well that extends from 0 to a in the x direction, and from 0 to b in the y direction. a) Write down the time independent wave function Psi(x, y) for an electron inside this 2D infinite square well in terms of n_x and n_y, the quantum numbers corresponding to the x and y part of the wave function, respectively. b) Calculate the energy of the first excited state. Assume that a = 0.5 nm...
In solving the particle in a one dimensional infinite depth box problem (0k x < a) we started with the function following is a true statement? (a) The value of k is found by requiring that the solution be normalized. (b) The function wx) is not an eigenfunciton of the operator d2/dx2 (c) It is necessary that this function equals a when x=0 (ie, Ψ(0) = a). (d) The boundary condition at x = 0 is used to show that...
3. For a particle moving in an infinite, one-dimensional, symmetric square well of width 2a, show that the (normalized) wave functions are of the form ?-kx).va. cos?x): "-1. 3.5 ,.. COS ? -?? r")(x)=?sin n-r | ; n-2, 4, 6 Express the state ?(x)=N sin,(rx/a) as a linear superposition eigenstates, and find its normalization constant N. of the above HINT sin39-3sin ?-4sin'?
3 At a given time, the normalised wave function for a particle in a one-dimensional infinite square well -a < x < a is given by 2 sin2 V inside the well and zero outside. Find the probability that a measurement of energy yields the eigenvalue En. (Hint: use data on page 6.) [6] Useful Data and Formulas = 1.60 x 10-19 C Elementary charge e h/2T=1.05 x 10-34 Js Planck's constant 3.00 x 108 m s-1 Speed of light...
An electron in a one-dimensional infinite potential well of width L is found to have the normalized wave function ψ(x)- sin(2 r ). (a) What is the probability of finding the electron within the interval from x=010 x = L/2 ? (b) At what position or positions is the electron most likely to be found? In other words, find the value(s) of x where the probability of finding the particle is the greatest?
onfined to move say, a typical size of a desk). Based on the nics, what is the minimum speed of the student? quantum (c). What should the quantum number n be if the student is moving with a speed of 1.25 m/s? (d)What is the separation of the energy levels of the student moving with this speed (extra points: 10 points)The time independent Schrodinger equation in the infinite 5. potential well is d2) 2m Eψ 0 . + dx2 h2...
Q4. Consider the 1D infinite square-well potential shown in the figure below. V(x) O0 Position (a) State the time-independent Schrödinger equation within the region 0<x<L for a particle with positive energy E 2 marks] (b) The wavefunction for 0<x< L can be written in the general form y(x) = Asin kx + B cos kx. Show that the normalised wavefunction for the 1D infinite potential well becomes 2sn'n? ?snT/where ( "1,2,3 ! where ( n = 1,2,5, ). [4 marks]...
quantum mechanics
Consider a particle confined in two-dimensional box with infinite walls at x 0, L;y 0, L. the doubly degenerate eigenstates are: Ιψη, p (x,y))-2sinnLx sinpry for 0 < x, y < L elsewhere and their eigenenergies are: n + p, n, p where n, p-1,2, 3,.... Calculate the energy of the first excited state up to the first order in perturbation theory due to the addition of: 2 2
Consider a particle confined in two-dimensional box with infinite...
(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...
Consider an 2D infinite square well that extends from 0 to a in the x direction, and from 0 to b in the y direction. a) Write down the time independent wave function Psi(x, y) for an electron inside this 2D infinite square well in terms of n_x and n_y, the quantum numbers corresponding to the x and y part of the wave function, respectively. b) Calculate the energy of the first excited state. Assume that a = 0.5 nm...
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