Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 8.1, Problem 8.2P
To determine
The best bound on for the one-dimensional harmonic oscillator using a trial wave function of the form
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a simple harmonic oscillator of mass m and angular frequency w is perturbed by additional potential bx3 ,evaluate the second order correction t the ground state energy of the oscillator.
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Introduction To Quantum Mechanics
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- At the microscopic level, it is possible to understand the spontaneous emission of an alpha particle as having that particle trapped into a potential well, and having it exiting through a tunnel effect. For the following, we will assume we have a potential which is in the following form: ⚫ The potential is symmetrical on the x-axis ⚫ The potential has also a cylindrical symmetry, i.e. in a polar base it does not depend on the polar angle ⚫ For any x =L, the value of the potential is supposed to be Ve**, with A being a positive value. 1) Solve the wave function of the system 2) Assuming you have a probability P to observe the alpha decay, can you write down the equation to compute the values of the coefficients A of the wave functions?arrow_forwardConsider a weakly anharmonic a 1D oscillator with the poten- tial energy m U(x) = w?a² + Ba* 2 Calculate the energy levels in the first order in the small anharmonicity parameter 3 using TIPT and the ladder operators.arrow_forwardBy taking the derivative of the first equation with respect to b, show that the second equation is true. Use this result to determine △x and △p for the ground state of the simple harmonic oscialltor.arrow_forward
- The wave function for the first excited state y, for the simple harmonic oscillator is y, = Axe (ax-/2), Normalize the wave function to find the value of the constant A. (Use the following as necessary: a) A%3D Determine (x), (x-), and y (x2) - (x)². (Use the following as necessary: a) (x) (x?) V (x?) - (x)? : Need Help? Read Itarrow_forwardConsider the scattering of a particle by a regular lattice of basis a, b, c. The interaction with the lattice can be written as V = E, V(Ir – r,|). where V (r-r,|) is the potential of each atom and is spherically symmetric about the atom's lattice point. Show using the Born approximation that the condition for non-vanishing scattering is that the Bragg law be satisfied.arrow_forward6QM Please answer question throughly and detailed.arrow_forward
- The odd parity eigenstates of the infinte square well , with potential V = 0 in the range −L/2 ≤ x ≤ L/2, are given by : (see figure) and have Ψn(x, t) = 0 elsewhere , for n=2 , 4 , 6 etc I have got the expectation value of momentum for ⟨p⟩ and ⟨p 2⟩ for n = 2 (see figures) Determine the uncertainty in momentum, ∆p, for a particle with n = 2, and use your result to put a lower bound on the uncertainty in position via Heisenberg’s uncertainty relation.arrow_forwardConsider as a model of a hydrogen atom a particle trapped in a one-dimensional, infi nite potential well of width 2a0 (the ground-state hydrogen atom’s diameter). Find the electron’s ground-state energy and comment on the resultarrow_forwardA measurement of the magnitude of Sx for the spin 1/2 system produces an eigenvalue I FIN 2 which is then followed by a measurement of Sx cos Ø+ Sy sin Ø Determine the probability of finding the state with the result EIN 2 of the second measurement.arrow_forward
- 40. The first excited state of the harmonic oscillator has a wave function of the form y(x) = Axe-ax². (a) Follow thearrow_forward1arrow_forwardA particle of mass m is trapped in a three-dimensional rectangular potential well with sides of length L, L/ √2, and 2L. Inside the box V = 0, outside V = ∞. Assume that Ψ = Asin (k1x) sin (k2y) sin (k3z) inside the well. Substitute this wave function into the Schrödinger equation and apply appropriate boundary conditions to find the allowed energy levels. Find the energy of the ground state and first four excited levels. Which of these levels are degenerate?arrow_forward
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