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- (a) For a particle in the stationary state n of a one dimensional box of length a, find the probability that the particle is in the region 0 x a/4. (b) Calculate this probability for n = 1, 2, and 3 Sketch and | |2 for the n = 4 and n = 5 states of a particle in a one-dimensional box.For a particle in the stationary state n of a one dimensional box of length a, find the probability that the particle is in the region 0 xa/4. (b) Calculate this probability for n = 1, 2, and 3A glow worm of mass 5.0 g emits red light (650nm) with a power of 0.10W entirely in the backward direction. To what speed will it have accelerated after 10y if released into free space and assumed to live? Energy emitted = P(W)-At(s). Assume the conservation of linear momentum, which means that the loss of a photon imparts an equivalent momentum in the opposite direction. total energy The total momentum imparted to the glow-worm: p = Npphoton where energy/photon x= 650nm N = Energy emitted p = mv Energyphoton V= = { PglowwormN Pphoton N = # of photons = Pphoton = momentum/photon 5g=00005kg = (mv) glowworm At =10yrs => At-see Total Energy = Pat energy/photon = (E=hr) = hec Pphoton = 12 숫
- For a particle in a two-dimensional box with sides of equal lengths, draw rough sketches of contours of constant || for the states (a) nx=2, ny=1; (b) nx =2, ny=2. At what points in the box is || a maximum for each state?A 1.00 kg weight is traveling at a constant velocity of 10.0 ms-1, in a cubic room 3.00 m on an edge.(a). Determine the sum of the squares of its three quantum numbers.(b). What would be the ground state energy for this weight?Find the uncertainty ox in x as a function of the uncertainties ou and oy in u and v for the following functions: (u+v) (а) х %3 2 (b) x =uv? (c) x= u? +v?
- 5) Richard Feynman called the Euler relation the most remarkable formula in mathematics. Use the Euler relation to give the value of the following quantities. eio=1 ein/2_ ein = ei2π = (Hix = (rcosx = i sin x)[p* L₂ da 0 HO H + →Consider a hydrogen atom in the 1s state. (a) For what value of r is the potential energy U(r) equal to the total energy E? Express your answer in terms of a. This value of r is called the classical turning point, since this is where a Newtonian particle would stop its motion and reverse direction. (b) For r greater than the classical turning point, U(r)7 E. Classically, the particle cannot be in this region, since the kinetic energy cannot be negative. Calculate the probability of the electron being found in this classically forbidden region.
- Consider a particle of mass m confined to a one-dimensional box of length L and in a state with normalized wavefunction ψn. (a) Without evaluating any integrals, explain why ⟨x⟩ = L/2. (b) Without evaluating any integrals, explain why ⟨px⟩ = 0. (c) Derive an expression for ⟨x2⟩ (the necessary integrals will be found in the Resource section). (d) For a particle in a box the energy is given by En = n2h2/8mL2 and, because the potential energy is zero, all of this energy is kinetic. Use this observation and, without evaluating any integrals, explain why <p2x> = n2h2/4L2.10. A particle of mass m is confined to move in one dimension on the domain 0 < x < ∞, and its quantum state is associated with the wavefunction (x) = Næe-ar where N is the normalization and a is a constant having units of inverse length. Normalize the wavefunction and derive an expression for (1/2) for the particle. Answer ²6° N² 2! (2a)³ x²2e-2ax dx = N², N = 2a³/2 = N² 4a³ = 1The components of angular momentum are as follows: I. L, = yp: - -Py II. L. = xp, - YP; %3D %3D | I. L, 3 ур. — Ер, IV. L, %3D хp. - ур. Which of the components are correct? (A) I and III (В) II and IV (C) I and II (D) All are correct