Problem 15. In the case of a particle tunneling through the barrier, we made equal both functions and derivatives at the boundary between region I and II, and between II and III, see eq. (9a – 9d). But in the case of the particle in the box, we made both functions equal at x = 0 and at x = L. Explain why we did not consider derivatives in this case.
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- For a single particle in 1D, which of the following cannot be found exactly using initial conditions, i.e., deterministically? The position at time t using Newton's 2nd equation. O The momentum at time t using Hamilton's equations. The wave function ) at time t using Schrodinger equation. O The kinetic energy at time t using Schrodinger equation.Infinite/finite Potential Well 1. Sketch the solution (Wave function - Y) for the infinite potential well and show the following: (a) Specify the boundary conditions for region I, region II, and region III. (i.e. U = ?, and x = ?) (b) Specify the length of the potential well (L=10 cm) (c) Which region will have the highest probability of finding the particle?Considering the problem of a time independent one- dimensional particle in a box with a dimension from 0 to 2a. From the quantum point of view Find the following: 1. The allowed energy levels for this particle. 2. The normalized wave function that describes this particle.
- Consider a quantum particle with energy E approaching a potential barrier of width L and heightV0 > E from the right (as shown in image). The wavefunction of the particles in the region x > L is given by ψ = A exp {−i (kx + ωt)} , where A, k and ω are all constants. Use the Gamow factor formalism to calculate an approximate expression for the transmission rate of these particles through the barrier.An electron has a kinetic energy of 13.3 eV. The electron is incident upon a rectangular barrier of height 21.5 eV and width 1.00 nm. If the electron absorbed all the energy of a photon of green light (with wavelength 546 nm) at the instant it reached the barrier, by what factor would the electron's probability of tunneling through the barrier increase?a question of quantum mechanics: Consider a particle in a two-dimensional potential as shown in the picture Suppose the particle is in the ground state. If we measure the position of the particle, what isthe probability of detecting it in region 0<=x,y<=L/2 ?
- Evaluate the E expressions for both the Classical (continuous, involves integration) and the Quantum (discrete, involves summation) models for the energy density u, (v).At time t = 0 the normalized wave function for a particle of mass m in the one-dimensional infinite well (see first image) is given by the function in the second image. Find ψ(x, t). What is the probability that a measurement of the energy at time t will yield the result ħ2π2/2mL2? Find <E> for the particle at time t. (Hint: <E> can be obtained by inspection, without an integral)