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Problem 2 (20 pts): a) (10 pts) The wavefunction given below corresponds to a confined particle....
P7B.8 A normalized wavefunction for a particle confined between 0 and L in the x direction,...
P7B.8 A normalized wavefunction for a particle confined between 0 and L in the x direction, and between 0 and L in the y direction (that is, to a square of side L) is Ψ= (2/L) sin(nx/L) sin(ny/L). The probability of finding the particle between x, and x, along x, and between y, and y, along y is P- Calculate the probability that the particle is: (a) between 0 and x L/2,y O and y L/2 (i.e, in the bottom...
7. Compare the functional form of the wavefunction derived for a particle confined to a 2D...
7. Compare the functional form of the wavefunction derived for a particle confined to a 2D plane, Y., , to one confined to a 1D line, Y., or Y... Does there appear to be a general relationship between the two solutions? If so, specifically identify how they are related. Model 2 The following are the normalized solutions to the Schrödinger equations for a particle confined to a plane defined in Model 1. V., ,(x,y) = Pnevy »* Vaba sin2= sin2,...
(b) Given that a particle is restricted to the region 065L < x normalized wavefunction, proportional to 0.67L, in...
(b) Given that a particle is restricted to the region 065L < x normalized wavefunction, proportional to 0.67L, in a box of length L and has a sin(nm/L) n=1,2, Show that the probability P, of finding the particle within the two regions when n applying both the integral and approximation method. 1 is similar, b Note: sin2x (1-cos2x)/2
(b) Given that a particle is restricted to the region 065L
Problem 3 (10 pts) The wavefunction of a particle in an infinite potential well, of width...
Problem 3 (10 pts) The wavefunction of a particle in an infinite potential well, of width a, is initially given by 16 ?(x, t-0) sin"(? x/a) cos(nx/a) Find the expression for ?(x, t) for all t > 0
Problem 4 Suppose we know that a particle of mass is stuck on the x-axis, confined...
Problem 4 Suppose we know that a particle of mass is stuck on the x-axis, confined to the region -1<x< 1. Its wavefunction is given -x) -1 << < 1 < -1 or 2 > 1 where A is a real, as-yet-undetermined constant. We'll assume that all numbers are in Sl units, without actually writing the units down. a) Draw a set of graph axes below like the one below and draw a sketch of this wavefunction on the axes....
Particle in a box Figure 1 is an illustration of the concept of a particle in...
Particle in a box Figure 1 is an illustration of the concept of a particle in a box. V=00 V=00 V=0 Figure 1. A representation of a particle in a box, where the potential energy, V, is zero between x = 0 and x = L and rises abruptly to infinity at the walls. The Schrödinger equation for a particle in a box reads t² d²u Y +V(x)y = Ey 2m dx2 + (1) where ħ=h/21 , y represents the...
The wave function given below is suggested to fit the particle in a box of length...
The wave function given below is suggested to fit the particle in a box of length L in one dimension: Duh!! also known as the particle on a line: V=N (L x-); where N is the normalization constant. Problem One. List three conditions (in a short phrase) that make any wavefunction acceptable. For each condition, show that the above wavefunction satisfies the condition you listed. (Use the allotted spaces below to answer the question). (1) (III).
8. Compare the expression derived for translational energy for a particle confined to a 2D plane,...
8. Compare the expression derived for translational energy for a particle confined to a 2D plane, Em..., to one confined to a 1D line, E, or E. Does there appear to be a general relationship between the two solutions? If so, specifically identify how they are related. Model 2 The following are the normalized solutions to the Schrödinger equations for a particle confined to a plane defined in Model 1. V.,„, (x,y)= \sin ",7 sin ", Ty E. (x,y) =...
2.2 Two-level system A particle in the box is described by the following wavefunction 1 1...
2.2 Two-level system A particle in the box is described by the following wavefunction 1 1 V(x, t) + V2 V2 = Um(x)e -i(Em/h) In other words, this state is a superposition of two modes: n-th, and m-th. A superposition that involves only two modes (not necessarily particle in the box modes, but any two modes) is called a "two-level system”. A more modern name for such a superposition is a "qubit”. a) Come up with an expression for the...
Problem 10 (Problem 2.24 in textbook) The wavefunction for the electron in a hydrogen atom in its...
Problem 10 (Problem 2.24 in textbook) The wavefunction for the electron in a hydrogen atom in its ground state (the 1s state for which n 0, l-0, and m-0) is spherically symmetric as shown in Fig. 2.14. For this state the wavefunction is real and is given by exp-r/ao h2Eo 5.29 x 10-11 m. This quantity is the radius of the first Bohr orbit for hydrogen (see next chapter). Because of the spherical symmetry of ịpo, dV in Eq. (2.56)...
P7B.8 A normalized wavefunction for a particle confined between 0 and L in the x direction, and between 0 and L in the y direction (that is, to a square of side L) is Ψ= (2/L) sin(nx/L) sin(ny/L). The probability of finding the particle between x, and x, along x, and between y, and y, along y is P- Calculate the probability that the particle is: (a) between 0 and x L/2,y O and y L/2 (i.e, in the bottom...
7. Compare the functional form of the wavefunction derived for a particle confined to a 2D plane, Y., , to one confined to a 1D line, Y., or Y... Does there appear to be a general relationship between the two solutions? If so, specifically identify how they are related. Model 2 The following are the normalized solutions to the Schrödinger equations for a particle confined to a plane defined in Model 1. V., ,(x,y) = Pnevy »* Vaba sin2= sin2,...
(b) Given that a particle is restricted to the region 065L < x normalized wavefunction, proportional to 0.67L, in a box of length L and has a sin(nm/L) n=1,2, Show that the probability P, of finding the particle within the two regions when n applying both the integral and approximation method. 1 is similar, b Note: sin2x (1-cos2x)/2
(b) Given that a particle is restricted to the region 065L
Problem 3 (10 pts) The wavefunction of a particle in an infinite potential well, of width a, is initially given by 16 ?(x, t-0) sin"(? x/a) cos(nx/a) Find the expression for ?(x, t) for all t > 0
Problem 4 Suppose we know that a particle of mass is stuck on the x-axis, confined to the region -1<x< 1. Its wavefunction is given -x) -1 << < 1 < -1 or 2 > 1 where A is a real, as-yet-undetermined constant. We'll assume that all numbers are in Sl units, without actually writing the units down. a) Draw a set of graph axes below like the one below and draw a sketch of this wavefunction on the axes....
Particle in a box Figure 1 is an illustration of the concept of a particle in a box. V=00 V=00 V=0 Figure 1. A representation of a particle in a box, where the potential energy, V, is zero between x = 0 and x = L and rises abruptly to infinity at the walls. The Schrödinger equation for a particle in a box reads t² d²u Y +V(x)y = Ey 2m dx2 + (1) where ħ=h/21 , y represents the...
The wave function given below is suggested to fit the particle in a box of length L in one dimension: Duh!! also known as the particle on a line: V=N (L x-); where N is the normalization constant. Problem One. List three conditions (in a short phrase) that make any wavefunction acceptable. For each condition, show that the above wavefunction satisfies the condition you listed. (Use the allotted spaces below to answer the question). (1) (III).
8. Compare the expression derived for translational energy for a particle confined to a 2D plane, Em..., to one confined to a 1D line, E, or E. Does there appear to be a general relationship between the two solutions? If so, specifically identify how they are related. Model 2 The following are the normalized solutions to the Schrödinger equations for a particle confined to a plane defined in Model 1. V.,„, (x,y)= \sin ",7 sin ", Ty E. (x,y) =...
2.2 Two-level system A particle in the box is described by the following wavefunction 1 1 V(x, t) + V2 V2 = Um(x)e -i(Em/h) In other words, this state is a superposition of two modes: n-th, and m-th. A superposition that involves only two modes (not necessarily particle in the box modes, but any two modes) is called a "two-level system”. A more modern name for such a superposition is a "qubit”. a) Come up with an expression for the...
Problem 10 (Problem 2.24 in textbook) The wavefunction for the electron in a hydrogen atom in its ground state (the 1s state for which n 0, l-0, and m-0) is spherically symmetric as shown in Fig. 2.14. For this state the wavefunction is real and is given by exp-r/ao h2Eo 5.29 x 10-11 m. This quantity is the radius of the first Bohr orbit for hydrogen (see next chapter). Because of the spherical symmetry of ịpo, dV in Eq. (2.56)...
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