3. A particle of mass moves in one dimension in a potential given by v(s)-c8(s). where 8() is the Dirac delta function. The particle is bound. Find the value, such that the probability of finding the particle with
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A: Solution by image is shown belowExplanation:Step 1: Step 2: Step 3: Step 4:
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- 3. A system of N harmonic oscillators of frequency w are prepared in identical initial states of wavefunction Þ(x,0). It is found that the measurement of the energy of the system at t = 0 gives 0.5hw with probability 0.25, 1.5hw with probability 0.5 and 2.5hw with probability 0.25. a. Write a possible function p(x, 0). b. Write the corresponding (x, t). c. What is the expectation value of the Hamiltonian in the state p(x,t) ? d. Calculate the expectation value of position at time tQ1. Consider the finite square well potential shown in the following diagram: U(x) E>0 L The potential is given by: for xL| -U. for 0 0is incident on this region from the left. Using the plane A particle with energy wave approximation for the particle: a) Show that Y = Ae*+Be¬k* is a suitable general solution to the time-independent Schrödinger wave-equation (TISE) that applies in the region x L write down the four equations arising from the boundary conditions that apply at x=0 and x=L .n=2 35 L FIGURE 1.0 1. FIGURE 1.0 shows a particle of mass m moves in x-axis with the following potential: V(x) = { 0, for 01. Consider a one-dimensional rectangular potential barrier V(x): z(0, r)a 0(r(a V(x) - (1) where Vo is positive. A particle with energy E < V, approaches the barrier from the left (see figure Fig. 1). 2 E 1 3 x=0 x=a Figure 1 i Find in each region the wave function by solving the Schrodinger equation. ii By definition, the transmission coefficient T is defined as the transmitted current divided by the incident current: T- İranamtted Jimetdent where j 2mi (2) evaluate T in terms of the amplitudes of transmitted and incident wave functions. iii Using the boundary conditions prove that: T = where k2 1+E Ninh? kga is the wave vector in region 2.4. A simple model of a radioactive nuclear decay assumes that alpha particles are trapped inside a nuclear potential well. An alpha particle is a particle made out of two protons and two neutrons and has a mass of 3.73 GeV/c². The nuclear potential can be modeled as a pair of barriers each with a width of 2.0 fm and a height of 30.0 MeV. Find the probability for an alpha particle to tunnel across one of the potential barriers if it has a kinetic energy of 20.0 MeV.2. Consider a particle of mass m in an infinite square well, 0(≤ x ≤a). At the time t = 0 the particle is in the ground (n = 1). Then at t> 0 a weak time-dependent external potential is turned on: H' = Axe T24. Consider a modified box potential with V(x) = V₁x, Vi(ar), x a Use the orthogonal trial function = c₁f₁+c₂f₂ with f₁ = √√sin (H) and f2 = √√ √√sin sin (2) to determine the upper bound to ground state energy.0 A physical system is described by a two-dimensional vector space with Hamiltonian operator Ĥ given by Ĥ = (_) where a is a constant. At time t = 0, the system is prepared in state (t = 0)) = -i2.5 0 determine the expectation value (Ŝ) at time t = πħ/(4x). O a. 2.17 O b. -2.50 O c. -1.25 O d. 2.50 O e. 5.00 0 (¹). For operator $ = (2 i2.5