Problem 4. 1. Find the energy and the wave function for a particle moving in an infinite spherical well of radius a with l = 0. Hint: Replace (r) → f(r)/r.
Q: *Problem 2.22 The gaussian wave packet. A free particle has the initial wave function (x, 0) =…
A: Step 1: Gaussian wave packet: The initial wave function for a free particle is given as: ψ(x,0)…
Q: Problem 3.10 Is the ground state of the infinite square well an eigenfunction of momentum? If so,…
A: Introduction: The wave function of the stationary states of the infinite square well is given by:…
Q: Problem #1 (Problem 5.3 in book). Come up with a function for A (the Helmholtz free energy) and…
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Q: Problem 1.17 A particle is represented (at time=0) by the wave function A(a²-x²). if-a ≤x≤+a. 0,…
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Q: *Problem 1.14 A particle of mass m is in the state V (x, t) = Ae-a[(mx²/h)+it]_ where A and a are…
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Q: A particle confined in an infinite square well between x = 0 and r = L is prepared with wave…
A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…
Q: 5.1. (a) Prove that the parity operator is Hermitian. (b) Show that the eigenfunctions of the parity…
A: In quantum mechanics, the parity operator is a fundamental operator that reverses the spatial…
Q: How do I prove that the transition is successful in problem 7.22?
A: The given wave function is
Q: A particle moving in one dimension is in a stationary state whose wave function x a where A and a…
A: Note: As per the policy, only the first three subparts of the question will be solved. If you want…
Q: Q1. Consider the finite square well potential shown in the following diagram: U(x) E> 0 L х -U, The…
A: Let's first write the wave equations in the three regions ψI = Aeikx + B e-ikx…
Q: 4.8. The energy eigenfunctions V1, V2, V3, and 4 corresponding to the four lowest energy states for…
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Q: 4.8 a. Assuming that the Hamiltonian is invariant under time reversal, prove that the wave function…
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Q: Problem 3.27 Sequential measurements. An operator Ä, representing observ- able A, has two normalized…
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Q: SECTION B Answer TWO questions from Section B Question B1 Consider a finite potential step as shown…
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Q: A neutron of mass m with energy E a,V(x) =+Vo . I. Write down the Schrödinger equation for: region I…
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Q: If V0 = 4 eV, E = 1 eV and L = 0.01 nm, determine the probability of a quantum-mechanical electron…
A: Given a potential barrier with height V0=4 eV and barrier length L=0.01 nm and the energy of the…
Q: Consider a classical particle moving in a one-dimensional potential well u(x), as shown in Figure…
A: According to question, particle is in inside the one dimensional potential well of potential u(x),…
Q: We've looked at the wavefunction for a particle in a box. Soon we will look at other systems with…
A: Given that: ψ(ϕ)=12πeimϕEk^=-h22Id2dϕ2V=0
Q: 7.2 Consider an anisotropic harmonic oscillator described by the Hamiltonian 1 Pi + p; + p?) +…
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Q: 5.1 Consider a one-dimensional bound particle. Show that if the particle is in a stationary state at…
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Q: 4.7 a. Let y(x.t) be the wave function of a spinless particle corresponding to a plane wave in three…
A: Solution:-a). ψ(x,t)=expi(k.x-wt) ω*(x,-t)=exp-i(k.x+wt) ψ*x,t=expi-k.x-wt…
Q: E Assume an electron is initially at the ground state of a l-D infinite square well and is exposed…
A: Here we will use time dependent perturbation theory. Let us first find out the matrix coefficient…
Q: 2.1 Consider a linear chain in which alternate ions have masses M₁ and M2, and only nearest…
A: We have given a two dimensions linear lattice with lattice constant a/2 we have to find out the…
Q: Problem 2.7 A particle in the infinite square well has the initial wave function 0 < x < a/2, JA (a…
A: Quantum mechanics shows the nature of the motion of a microscopic particle. The behaviour of quantum…
Q: Problem 2.3 Show that there is no acceptable solution to the (time-independent) Schrödinger equation…
A: Solution:- From the Schrodinger equation for an infinite square will we…
Q: (e) Show that at time t = 4ma² /nħ, the wavefunction returns to its initial state. (f) Suppose the…
A: It is required to solve parts e and f as per the request.
Q: Question 2 2.1 Consider an infinite well for which the bottom is not flat, as sketched here. If the…
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Q: Consider a classical particle moving in a one-dimensional potential well u(x), as shown in Figure…
A: Solution: (A model of thermal expansion.) (a) Let the "system” be the particle's position and the…
Q: (a) Find Ao for the 1D function V(x) : Aoe-ik-xá
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Q: A particle is confined betweek = 7). Evaluate the probability to find the particle in an interval of…
A: Given: The length of the rigid wall is L=0.189 nm. The state of the particle is n=7. The…
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- 6.2. Solve the three-dimensional harmonic oscillator for which 1 V(r) = -— mw² (x² + y²² +2²) 2 by separation of variables in Cartesian coordinates. Assume that the one-dimensional oscillator has eigenfunctions (x) with corresponding energy eigenvalues En = (n + 1/2)hw. What is the degeneracy of the first excited state of the oscillator?2. A simple harmonic oscillator is in the state 4 = N(Yo + λ 4₁) where λ is a real parameter, and to and ₁ are the first two orthonormal stationary states. (a) Determine the normalization constant N in terms of λ. (b) Using raising and lowering operators (see Griffiths 2.69), calculate the uncertainty Ax in terms of .Problem 4.25 If electron, radius [4.138] 4πεmc2 What would be the velocity of a point on the "equator" in m /s if it were a classical solid sphere with a given angular momentum of (1/2) h? (The classical electron radius, re, is obtained by assuming that the mass of the electron can be attributed to the energy stored in its electric field with the help of Einstein's formula E = mc2). Does this model make sense? (In fact, the experimentally determined radius of the electron is much smaller than re, making this problem worse).
- 2.4. A particle moves in an infinite cubic potential well described by: V (x1, x2) = {00 12= if 0 ≤ x1, x2 a otherwise 1/2(+1) (a) Write down the exact energy and wave-function of the ground state. (2) (b) Write down the exact energy and wavefunction of the first excited states and specify their degeneracies. Now add the following perturbation to the infinite cubic well: H' = 18(x₁-x2) (c) Calculate the ground state energy to the first order correction. (5) (d) Calculate the energy of the first order correction to the first excited degenerated state. (3) (e) Calculate the energy of the first order correction to the second non-degenerate excited state. (3) (f) Use degenerate perturbation theory to determine the first-order correction to the two initially degenerate eigenvalues (energies). (3)Problem 4.45 What is the probability that an electron in the ground state of hydro- gen will be found inside the nucleus? (a) First calculate the exact answer, assuming the wave function (Equation 4.80) is correct all the way down to r = 0. Let b be the radius of the nucleus. (b) Expand your result as a power series in the small number € = 2b/a, and show that the lowest-order term is the cubic: P≈ (4/3)(b/a)³. This should be a suitable approximation, provided that bConsider a classical particle moving in a one-dimensional potential well u(x), as shown in Figure 6.10 (attached). The particle is in thermal equilibrium with a reservoir at temperature so the probabilities of its various states are determined by Boltzmann statistics.