1. [20P] Suppose a free particle is represented by the plane wave function: Y(x,t) = Aexp[i(1.58102x-7.9110+)],...

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1. [20P] Suppose a free particle is represented by the plane wave function: Y(x,t) = Aexp[i(1.58102x-7.9110+)], where all numbers are in SI units. What are the particle's momentum, kinetic energy, and mass? [Note: In nonrelativistic quantum mechanics, we ignore rest energy, so the frequency of a free particle is related to kinetic energy, K, alone, i.e., by w = K.] 2. [20P] Suppose a particle on the x axis is described by the following wave function, at some instant in time (say, t = 0): A 4(x, 0) = x + b where b is a given length and A is to be determined. (a) Use the fact that the probability for the particle to be somewhere on the x axis is 1 to determine the value of A. Express the result in terms of b. Hint: The value of A is to be chosen such that probability density function (PDF) of the position coordinate x is: Px,4(x) = |Y(x, 0)|, i.e., such that the normaliza- tion factor of 4(x, 0) has the value N = 1. A wavefunction 4 with N = 1 is also referred to as a "normalized wave function". (b) Determine the value of x0 such that the probability of finding the particle between -xo and xo is equal to the probability of finding it outside this interval? Express the result in terms b. 3. [20P] A mosquito of mass 0.15 mg is found to be flying at a speed of 50 cm/s within an uncertainty of 0.5 mm/s (a) How precisely may its position be known? (b) Does this inherent uncertainty present any hindrance to the application of classical mechanics? Explain. 4. [20P] To how small a region must an electron be confined for borderline relativistic speedssay, 0.05c-to become reasonably likely? On the basis of this, would you expect relativistic effects to be prominent for hydrogen's electron, which has an orbit radius near 10-10 m? For a lead atom "inner-shell" electron of orbit radius 10-12 m? 5. [20P] The po is a subatomic particle of fleeting existence. Data tables don't usually quote its lifetime. Rather, they quote a "width", meaning energy uncertainty, of about 150 MeV. Roughly what is its lifetime? 6. [20P] In problem 3 you showed that the position uncertainty of a macroscopic object is so much smaller than its actual physical dimensions that applying the uncertainty principle would be absurd. In this problem we gain some idea of how small an object would have to be before quantum mechanics might rear its head. The density of alu- minum, 2.7103 kg/m, is typical of solids and liquids around us. Suppose we could narrow down the velocity of an aluminum sphere to within an uncertainty of 1 m per decade. How small would it have to be for its position uncertainty to be at least as large as 0.1% of its radius? 7. [20P] Suppose a particle is connected to a spring and undergoes one-dimensional mo- tion. (a) Write an expression for the total (kinetic plus potential) energy of the particle in terms of its position x, its mass m, its momentum p, and the force constant of the spring. (b) Now treat the particle as a wave. Assume that the product of the uncertainties in position and momentum is governed by an uncertainty relation AxAph. Also assume that because x is, on average, zero, the uncertainty Ax is roughly equal to a typical value of x. Similarly, assume that Ap = [p. Eliminate p in favor of x in the energy expression. (c) Find the minimum possible energy for the wave, and express it in terms of the vibrational frequency of a mass on a spring: 1 f= 2 V m 1. [20P] Suppose a free particle is represented by the plane wave function: Y(x,t) = Aexp[i(1.58102x-7.9110+)], where all numbers are in SI units. What are the particle's momentum, kinetic energy, and mass? [Note: In nonrelativistic quantum mechanics, we ignore rest energy, so the frequency of a free particle is related to kinetic energy, K, alone, i.e., by w = K.] 2. [20P] Suppose a particle on the x axis is described by the following wave function, at some instant in time (say, t = 0): A 4(x, 0) = x + b where b is a given length and A is to be determined. (a) Use the fact that the probability for the particle to be somewhere on the x axis is 1 to determine the value of A. Express the result in terms of b. Hint: The value of A is to be chosen such that probability density function (PDF) of the position coordinate x is: Px,4(x) = |Y(x, 0)|, i.e., such that the normaliza- tion factor of 4(x, 0) has the value N = 1. A wavefunction 4 with N = 1 is also referred to as a "normalized wave function". (b) Determine the value of x0 such that the probability of finding the particle between -xo and xo is equal to the probability of finding it outside this interval? Express the result in terms b. 3. [20P] A mosquito of mass 0.15 mg is found to be flying at a speed of 50 cm/s within an uncertainty of 0.5 mm/s (a) How precisely may its position be known? (b) Does this inherent uncertainty present any hindrance to the application of classical mechanics? Explain. 4. [20P] To how small a region must an electron be confined for borderline relativistic speedssay, 0.05c-to become reasonably likely? On the basis of this, would you expect relativistic effects to be prominent for hydrogen's electron, which has an orbit radius near 10-10 m? For a lead atom "inner-shell" electron of orbit radius 10-12 m? 5. [20P] The po is a subatomic particle of fleeting existence. Data tables don't usually quote its lifetime. Rather, they quote a "width", meaning energy uncertainty, of about 150 MeV. Roughly what is its lifetime? 6. [20P] In problem 3 you showed that the position uncertainty of a macroscopic object is so much smaller than its actual physical dimensions that applying the uncertainty principle would be absurd. In this problem we gain some idea of how small an object would have to be before quantum mechanics might rear its head. The density of alu- minum, 2.7103 kg/m, is typical of solids and liquids around us. Suppose we could narrow down the velocity of an aluminum sphere to within an uncertainty of 1 m per decade. How small would it have to be for its position uncertainty to be at least as large as 0.1% of its radius? 7. [20P] Suppose a particle is connected to a spring and undergoes one-dimensional mo- tion. (a) Write an expression for the total (kinetic plus potential) energy of the particle in terms of its position x, its mass m, its momentum p, and the force constant of the spring. (b) Now treat the particle as a wave. Assume that the product of the uncertainties in position and momentum is governed by an uncertainty relation AxAph. Also assume that because x is, on average, zero, the uncertainty Ax is roughly equal to a typical value of x. Similarly, assume that Ap = [p. Eliminate p in favor of x in the energy expression. (c) Find the minimum possible energy for the wave, and express it in terms of the vibrational frequency of a mass on a spring: 1 f= 2 V m