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Assume an infinitely deep potential well with V(z) 0 for 0 Sz S a and V(2)...
neutron is trapped in an infinitely deep potential well 2.0fm in width. Determine (a) the four...
neutron is trapped in an infinitely deep potential well 2.0fm in width. Determine (a) the four lowest possible energy states and (b) their wave functions. (c) What is the wavelength and energy of a photon emitted when the neutron makes a transition between the two lowest states? In what region of the EM spectrum does this photon lie?
A potential modified from and infinitely deep well is shown below. It is located at x...
A potential modified from and infinitely deep well is shown below. It is located at x = 0 and is narrow enough that the wave function does not change across the well. please not E2= -4U. 0 E1 -300 E2 1) Sketch the wave function for E1 and Ez ii) Write down the time independent Schrödinger equation for the regions x > 0 and x <0. Then find a general solution for each region. iii) Simplify the solutions using the...
An infinitely deep square well has width L 2.5 nm. The potential energy is V =...
An infinitely deep square well has width L 2.5 nm. The potential energy is V = 0 eV inside the well (i.e., for 0 s xs L) Seven electrons are trapped in the well. 1) What is the ground state (lowest) energy of this seven electron system? Eground eV Submit 2) What is the energy of the first excited state of the system? NOTE: The first excited state is the one that has the lowest energy that is larger than...
Consider a particle in a 1-dimensional ininite square well potential {0, V(z)=Í oo, (-a < z...
Consider a particle in a 1-dimensional ininite square well potential {0, V(z)=Í oo, (-a < z <a) elsewhere The particle is initially localized in the right side of the well (O S a) Calculate the probability that at later times, an energy measurement will yield the energy of the first excited state of this system
Consider a particle in a 1-d well with potential V(x) =-U for-d < x < d,...
Consider a particle in a 1-d well with potential V(x) =-U for-d < x < d, and V(z) 0 elsewhere. We will use the variational wave function v(z) = A(b + r), t(x)-A(b-x), -b < r < 0, 0 < x < b, to show that a bound state exists for any U0. a) Normalize the wave function. Find the expectation values of the kinetic and potential energies b) Show that for sufficiently large b, with b> d, the expectation...
[12 6. Consid er a particle of mass m moving in an infinitely deep square well...
[12 6. Consid er a particle of mass m moving in an infinitely deep square well potential of width a, whose wave function at time t 0 is where on Ce) is the normaized wave function of the n-th eigenstate of the Hamitonian of that particle The corresponding eigen-energy of the n-th state is 2ma?n 1,2,3,... (e) Find the average energy of the system (ie. the expectation value () (b) Write down the wave function p(z,t) at a later time...
1l] A particle with mass m and energy E is inside a square tube with infinite potential barriers at x-o, x-a, y 0,...
1l] A particle with mass m and energy E is inside a square tube with infinite potential barriers at x-o, x-a, y 0, y a. The tube is infinitely long in the +z-direction. (a) Solve the Schroedinger equation to derive the allowed wave functions for this particle. Do not try to normalize the wave functions, but make sure they correspond to motion in +2-direction. (b) Determine the allowed energies for such a particle. (c) If we were to probe the...
4.) A spherical square well is a square well in three spatial dimensions which satisfies -Vo,...
4.) A spherical square well is a square well in three spatial dimensions which satisfies -Vo, ifr <a/2; (3) V (r) = 0, ifr>a/2, where r is the radial coordinate and a is a constant. a) Determine the ground-state solution to the effectively one-dimensional Schroedinger equation in r for a particle of mass m, as well as the transcendental equation the ground-state energy must satisfy. b) If you replace r by a, the one-dimensional coordinate, does your answer differ from...
3. Consider a free particle on a circle. That is, consider V(z) = 0 and wave...
3. Consider a free particle on a circle. That is, consider V(z) = 0 and wave functions Ψ(z, t) which are periodic functions of z: Ψ(z,t) = Ψ(z + L, t). a) Solve the Time-Independent Schroedinger equation. For each allowed energy, En, you will find two solutions, (s). Why does this not contradict the theorem that we proved in class about the non-degeneracy of the solutions to the TISE in one dimension? b) Start with the initial condition Ψ(z,0) sin2(nz/L)....
Consider a one-dimensional well with one impenetrable wall. The potential energy is given by 0 x...
Consider a one-dimensional well with one impenetrable wall. The potential energy is given by 0 x < 0 V(x) = { -V. 0 < x < a 10 x > a We showed in the homework that the allowed energies for the eigenstates of a bound particle (E < 0) in this potential well satisfy the transcendental function -cotĚ = 16 - 52 $2 where 5 = koa, and ko = V2m(Vo + E)/ħ, and 5o = av2mV /ħ (a)...
A potential modified from and infinitely deep well is shown below. It is located at x = 0 and is narrow enough that the wave function does not change across the well. please not E2= -4U. 0 E1 -300 E2 1) Sketch the wave function for E1 and Ez ii) Write down the time independent Schrödinger equation for the regions x > 0 and x <0. Then find a general solution for each region. iii) Simplify the solutions using the...
An infinitely deep square well has width L 2.5 nm. The potential energy is V = 0 eV inside the well (i.e., for 0 s xs L) Seven electrons are trapped in the well. 1) What is the ground state (lowest) energy of this seven electron system? Eground eV Submit 2) What is the energy of the first excited state of the system? NOTE: The first excited state is the one that has the lowest energy that is larger than...
Consider a particle in a 1-dimensional ininite square well potential {0, V(z)=Í oo, (-a < z <a) elsewhere The particle is initially localized in the right side of the well (O S a) Calculate the probability that at later times, an energy measurement will yield the energy of the first excited state of this system
Consider a particle in a 1-d well with potential V(x) =-U for-d < x < d, and V(z) 0 elsewhere. We will use the variational wave function v(z) = A(b + r), t(x)-A(b-x), -b < r < 0, 0 < x < b, to show that a bound state exists for any U0. a) Normalize the wave function. Find the expectation values of the kinetic and potential energies b) Show that for sufficiently large b, with b> d, the expectation...
[12 6. Consid er a particle of mass m moving in an infinitely deep square well potential of width a, whose wave function at time t 0 is where on Ce) is the normaized wave function of the n-th eigenstate of the Hamitonian of that particle The corresponding eigen-energy of the n-th state is 2ma?n 1,2,3,... (e) Find the average energy of the system (ie. the expectation value () (b) Write down the wave function p(z,t) at a later time...
1l] A particle with mass m and energy E is inside a square tube with infinite potential barriers at x-o, x-a, y 0, y a. The tube is infinitely long in the +z-direction. (a) Solve the Schroedinger equation to derive the allowed wave functions for this particle. Do not try to normalize the wave functions, but make sure they correspond to motion in +2-direction. (b) Determine the allowed energies for such a particle. (c) If we were to probe the...
4.) A spherical square well is a square well in three spatial dimensions which satisfies -Vo, ifr <a/2; (3) V (r) = 0, ifr>a/2, where r is the radial coordinate and a is a constant. a) Determine the ground-state solution to the effectively one-dimensional Schroedinger equation in r for a particle of mass m, as well as the transcendental equation the ground-state energy must satisfy. b) If you replace r by a, the one-dimensional coordinate, does your answer differ from...
3. Consider a free particle on a circle. That is, consider V(z) = 0 and wave functions Ψ(z, t) which are periodic functions of z: Ψ(z,t) = Ψ(z + L, t). a) Solve the Time-Independent Schroedinger equation. For each allowed energy, En, you will find two solutions, (s). Why does this not contradict the theorem that we proved in class about the non-degeneracy of the solutions to the TISE in one dimension? b) Start with the initial condition Ψ(z,0) sin2(nz/L)....
Consider a one-dimensional well with one impenetrable wall. The potential energy is given by 0 x < 0 V(x) = { -V. 0 < x < a 10 x > a We showed in the homework that the allowed energies for the eigenstates of a bound particle (E < 0) in this potential well satisfy the transcendental function -cotĚ = 16 - 52 $2 where 5 = koa, and ko = V2m(Vo + E)/ħ, and 5o = av2mV /ħ (a)...
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