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Exercises 3.6 Problems

1.

Probabilities with dice. This problem gets at the idea of a probability distribution. Take two of your dice, roll them, and record the sum of the spots showing on the top faces. (This should be a number between 2 and 12!) Repeat until you have recorded 20 results.

  1. Collect your results in three “bins.” How many trials do you find in the bin containing numbers 2, 3, 4, or 5? How many in the 6, 7, or 8 bin? In the 9, 10, 11, or 12 bin? Try to explain any patterns.

  2. In problem session, combine your results with others to estimate the probabilities of getting a roll in each of the three bins.

  3. Can you calculate theoretical probabilities? How do they compare with individual or class results?

2.

Classical probabilities for finding your car. John, an aspiring physics student, works part-time parking cars at a downtown hotel. The lot is a long, underground tunnel, with all the cars parked in a single long row, \(600\Xunits{m}\) long. When owners return for their cars, instead of telling them exactly where to find their cars, he describes the location in terms of probability and probability density.

  1. Mr. Vanderbilt is told that his car “could be anywhere in the lot,” which means that the probability density is constant. Calculate the value of this uniform probability density \(P(x)\) for Mr. Vanderbilt to find his car a distance \(x\) from one end of the lot. (Answer in units of probability/m.)

  2. Find the probability that Mr. Vanderbilt's car is in the first \(100\Xunits{m}\) of the lot.

  3. Mrs. Reeve is told that the probability density to find her car is a constant \(P_1\) from \(x = 0\) to \(x=200\Xunits{m}\text{,}\) and a second constant \mbox{\(P_2\) = \(P_1/3\)} in for \(x=200\) to \(x=600\Xunits{m}\text{.}\) Find the different constant probability densities \(P_1\) for \(0 \lt x \lt 200\Xunits{m}\) and \(P_2\) for \(200\Xunits{m} \lt x \lt 600\Xunits{m}\text{.}\)

  4. Based on your results from part (c), find the probability that Mrs. Reeve's car is in the first \(400\Xunits{m}\) of the lot.

3.

The probability amplitude \(\psi(x)\) for a certain particle to be at position \(x\) is

\begin{equation*} \psi(x) = \frac{\sqrt{x}}{a\sqrt{2}}, \hspace{0.2in}\text{ for \(0\leq x \leq a\). } \end{equation*}

  1. Explain what the quantity \(|\psi(x)|^2\) tells us about the particle.

  2. Calculate the probability that the particle is found between \(x=0\) and \(x = a\text{.}\)

  3. Calculate the probability that the particle is found anywhere else (i.e., not between \(x=0\) and \(x=a\)).

4.

Determine the probability of finding a particle between \(x = 1.0\) and \(x = 2.0\) for the following wavefunctions:

  1. \(\displaystyle \psi(x) = 0.10x+0.50\)

  2. \(\displaystyle \psi(x) = 0.10x+0.50i\)

5.

An electron in a hydrogen atom has a spread in position typically around \(\sigma_x = 5\times 10^{-11}\Xunits{m}\text{.}\)

  1. Use the Heisenberg uncertainty relation to find a lower bound on the spread in momentum, \(\sigma_{p_x}\text{,}\) for the electron.

  2. Now find a lower bound on the spread in velocity, \(\sigma_{v_x}\text{.}\)

6.

The Heisenberg uncertainty relation does not pose much of a limitation on our macroscopic scale. Consider a blow dart of mass \(2.5\Xunits{g}\text{.}\) Let's imagine a measurement of the dart's position with a precision limited to \(1\Xunits{ \mu m}\text{.}\) Determine the lower bound on the spread in velocity imposed by the Heisenberg uncertainty relation.

7.

Fabrication of nano-scale devices requires the ability to position atoms and molecules with very small spatial spread \(\sigma_x\text{.}\)

  1. Use Heisenberg's uncertainty principle to make a rough estimate of the precision by which a carbon atom (mass \(2.00 \times 10^{-26}\Xunits{kg}\) and radius \(70\Xunits{pm}\)) can be confined (i.e., determine the smallest \(\sigma_x\)) to keep its minimum kinetic energy below typical molecular binding energies of around \(1\Xunits{eV}\text{.}\) Would you expect the uncertainty principle to be a problem if you wanted to arrange carbon atoms in a nanotech device with a precision of around one-hundredth of the radius of a carbon atom?

  2. Repeat the calculations in part (a) for a proton (mass \(1.67 \times 10^{-27}\Xunits{kg}\)), also with \(\langle K\rangle\) below \(1\Xunits{eV}\text{.}\)

  3. Compare your result from (b) to the size of the nucleus of a carbon atom (\(2.7 \times 10^{-15}\Xunits{m}\)). What does this imply about a proton confined inside a carbon nucleus? I.e., what must be true for a proton to remain bound inside a carbon nucleus?

8.

Back in 1975, Gordon Moore proposed that the number of transistors per area on integrated circuits roughly doubles every two years. This principle ( “Moore's Law” ) has worked surprisingly well for 40 years now, with transistors introduced in 2012 as small as \(22\Xunits{nm}\) and techniques are continually being developed to make them even smaller. But Moore's Law will eventually fail due to limitations imposed by the uncertainty principle.

  1. Use the uncertainty principle to calculate the smallest spread \(\sigma_x\) for an electron such that its minimum kinetic energy (due to uncertainty) is below the work function (binding energy) of silicon, which is \(4.05\Xunits{eV}\text{.}\)

  2. Let's assume that the smallest possible transistor has an area 100 times the square of the \(\sigma_x\) that you calculated in part (a). Given the area (approximated as the square of \(22\Xunits{nm}\)) for the best transistors from 2012, if Moore's Law continues to hold into the future, roughly what year will transistors reach this quantum limit?

9.

Explain in a few sentences why classical physics (Newtonian mechanics and Electricity and Magnetism) predict that atoms with electrons orbiting around a nucleus are unstable and can't exist indefinitely.

10.

Use the uncertainty principle to estimate the minimum \(\langle K\rangle\) for

  1. an electron confined to a region of \(50\Xunits{pm}\) (roughly the radius of a hydrogen atom);

  2. a DNA molecule (\(1.0 \times 10^{-25}\Xunits{kg}\)) confined to the nucleus of a cell \(3.0\Xunits{ \mu m}\) radius; and

  3. a \(5.0\Xunits{mg}\) grain of sand in a pill box with width \(2.0\Xunits{cm}\text{.}\)

11.

Use the uncertainty principle to explain in 2 or 3 sentences

  1. why it is not possible for a confined particle to have zero kinetic energy; and

  2. why everyday-size objects (even something as small as a speck of dust) often seem to have zero kinetic energy, even when confined to a small region.

12.

Given the equation \(df/dx = 4.0\sin(0.25x)\text{,}\) test (by direct substitution) to determine if the following functions are solutions. If so, determine possible values of any constants.

  1. \(f(x) = Bx^2\text{,}\)

  2. \(f(x) = B\sin(kx)\text{,}\)

  3. \(f(x) = B\cos(kx)\text{.}\)

13.

Given the equation \(d^2f/dx^2 = 5x + 6\text{,}\) test (by direct substitution) to determine if the following functions are solutions. If so, determine possible values of any constants.

  1. \(f(x) = Ax^3+Bx^2+Cx\text{;}\)

  2. \(f(x) = A\sin(kx)\text{.}\)

14.

Schrödinger equation for a classically allowed situation. Consider a particle of mass \(m\) in a region in which the potential energy is constant, i.e., \(U(x)=U_0\text{,}\) and assume that the total energy of the particle \(E\) is greater than the potential energy, i.e., \(E>U_0\text{.}\) (This is the case for classically allowed motion.) To determine the wave function we must find a function \(\psi(x)\) that satisfies the one-dimensional Schrödinger equation

\begin{equation*} -\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} + U(x)\psi(x) = E\psi(x)\text{.} \end{equation*}

In this problem you will try three “guesses” for \(\psi(x)\) and see if they satisfy Schrödinger's equation. The three “guesses” are

  • \(\displaystyle \psi_1(x) = Ax^2\)

  • \(\displaystyle \psi_2(x) = B\sin(kx)\)

  • \(\psi_3(x) = Ce^{-\kappa x}\text{,}\)

where \(A\text{,}\) \(B\text{,}\) \(C\text{,}\) \(k\text{,}\) and \(\kappa\) are undetermined real constants.

  1. Rearrange Schrödinger's equation so that the second derivative \(d^2\psi/dx^2\) is alone on the left.

  2. Plug \(\psi_1(x)\) into Schrödinger's equation and see if there is any choice for the constant \(A\) that will make \(\psi_1(x)\) satisfy the equation for all values of \(x\text{.}\)

  3. Plug \(\psi_2(x)\) into Schrödinger's equation and see if there is any choice for the constants \(B\) and \(k\) that will make \(\psi_2(x)\) satisfy the equation for all values of \(x\) .

  4. Plug \(\psi_3(x)\) into Schrödinger's equation and see if there is any choice for the constants \(C\) and \(\kappa\) that will make \(\psi_3(x)\) satisfy the equation for all values of \(x\text{.}\)

  5. You should have found that \(\psi_2(x)\) can be a solution for the proper choice of \(k\text{.}\) Determine the wavelength of the oscillations in terms of \(\hbar\text{,}\) \(m\text{,}\) \(E\text{,}\) and \(U_0\text{.}\) (i.e., solve for \(k\) and remember from the waves unit that \(k=2\pi/\lambda\text{.}\)) Is your result consistent with that predicted from the de Broglie relationship? (Hint: \(E-U_0\) is the kinetic energy \(K = p^2/2m\text{.}\) Re-write things in terms of the momentum and the answer should drop into your lap.)

15.

Schrödinger equation for classically forbidden situation. Consider a particle of mass \(m\) in a region with a constant potential energy \(U_0\text{,}\) and assume that the total energy of the particle \(E\) is less than the potential energy, i.e., \(E\lt U_0\text{.}\) (This isn't possible for classical motion, but continue anyway.) To determine the wave function we must find a function \(\psi(x)\) that satisfies the one-dimensional Schrödinger equation

\begin{equation*} -\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} + U(x)\psi(x) = E\psi(x)\text{.} \end{equation*}

In this problem you will try three “guesses” for \(\psi(x)\) and see if they satisfy Schrödinger's equation. The three “guesses” are

  • \(\displaystyle \psi_1(x) = Ax^2\)

  • \(\displaystyle \psi_2(x) = B\sin(kx)\)

  • and \(\psi_3(x) = Ce^{-\kappa x}\text{,}\)

where \(A\text{,}\) \(B\text{,}\) \(C\text{,}\) \(k\text{,}\) and \(\kappa\) are undetermined real constants.

  1. Rearrange Schrödinger's equation so that the second derivative \(d^2\psi/dx^2\) is alone on the left.

  2. Plug \(\psi_1(x)\) into Schrödinger's equation and see if there is any choice for the constant \(A\) that will make \(\psi_1(x)\) satisfy the equation for all values of \(x\text{.}\)

  3. Plug \(\psi_2(x)\) into Schrödinger's equation and see if there is any choice for the constants \(B\) and \(k\) that will make \(\psi_2(x)\) satisfy the equation for all values of \(x\text{.}\)

  4. Plug \(\psi_3(x)\) into Schrödinger's equation and see if there is any choice for the constants \(C\) and \(\kappa\) that will make \(\psi_3(x)\) satisfy the equation for all values of \(x\text{.}\)

16.

A special case of the quantum harmonic oscillator is described by Schrödinger's equation of the form

\begin{equation*} -\frac{d^2\psi}{dx^2} + 4x^2 \psi = E\psi\text{.} \end{equation*}

Substitute the trial solution \(\psi = Ae^{-x^2}\) and determine the value of the energy \(E\text{.}\)

17.

The general case Schrödinger equation for the one-dimensional harmonic oscillator is

\begin{equation*} -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2}m\omega^2 x^2 \psi = E\psi\text{.} \end{equation*}

Substitute the trial solution \(\psi(x) = Ae^{-ax^2}\) (with \(a>0\)) into this equation and determine the constants \(a\) and \(E\text{.}\) This is the ground state of the oscillator.