← Back to Quantum Mechanics

Problems & Solutions in Quantum Mechanics

Selected problems from Problems & Solutions in Quantum Mechanics (Lim, ed.)—sourced from Wisconsin, Berkeley, and Columbia qualifying exams.

Problem 1001Wisconsin

Quantum Effects in the Macroscopic World

Problem Statement

Consider whether quantum effects are observable in everyday macroscopic situations.

(a) A 1-kg ball is constrained to move in a track and is attached to a light spring with force constant $k = 2\,\text{N/m}$, making a simple pendulum. What is the zero-point oscillation amplitude?

(b) A marble of mass $m = 10\,\text{g}$ rolls along a smooth track. It comes to the bottom of a hill of height $h = 10\,\text{cm}$ with a velocity $v = 100\,\text{cm/s}$. It does not have enough energy to get over the hill by classical means. What is the probability that it will tunnel through?

(c) A tennis ball bounces against a wall. Can one observe the diffraction of the tennis ball?

Step-by-Step Solution

Part (a): Zero-Point Oscillation Amplitude

For a quantum harmonic oscillator, the ground-state energy is $E_0 = \frac{1}{2}\hbar\omega$. The zero-point oscillation amplitude $A$ is found by equating the ground-state energy to the maximum potential energy:

$$\frac{1}{2}kA^2 = \frac{1}{2}\hbar\omega \quad\Rightarrow\quad A = \sqrt{\frac{\hbar}{m\omega}} = \sqrt{\frac{\hbar}{\sqrt{mk}}}$$

With $m = 1\,\text{kg}$, $k = 2\,\text{N/m}$, $\omega = \sqrt{k/m} = \sqrt{2}\,\text{rad/s}$:

$$A = \sqrt{\frac{1.055 \times 10^{-34}}{1 \times \sqrt{2}}} \approx 0.41 \times 10^{-17}\,\text{m}$$

This is far smaller than any atomic dimension—quantum zero-point motion is utterly undetectable at macroscopic scales.

Part (b): Tunneling Probability for the Marble

The tunneling probability through a potential barrier is approximately:

$$T \sim \exp\!\Bigl(-\frac{2}{\hbar}\int_0^d \sqrt{2m(V-E)}\,dx\Bigr)$$

The kinetic energy is $E = \frac{1}{2}mv^2 = \frac{1}{2}(0.01)(1)^2 = 0.005\,\text{J}$. The potential barrier height is $V = mgh = 0.01 \times 9.8 \times 0.1 = 0.0098\,\text{J}$. Estimating the barrier width as $d \approx h = 0.1\,\text{m}$:

$$\frac{2}{\hbar}\sqrt{2m(V-E)} \cdot d \approx \frac{2}{1.055\times10^{-34}}\sqrt{2(0.01)(0.0048)} \times 0.1 \approx 0.9 \times 10^{30}$$

Therefore $T \sim e^{-0.9\times10^{30}} \approx 0$. The probability is so absurdly small that tunneling of macroscopic objects is never observed.

Part (c): Diffraction of a Tennis Ball

The de Broglie wavelength of a tennis ball ($m \approx 0.05\,\text{kg}$, $v \approx 10\,\text{m/s}$):

$$\lambda = \frac{h}{mv} = \frac{6.63\times10^{-34}}{0.05 \times 10} = 1.3 \times 10^{-33}\,\text{m} = 1.3 \times 10^{-32}\,\text{cm}$$

This wavelength is unimaginably smaller than any physical aperture. Diffraction effects require $\lambda \sim d$ (aperture size), so no diffraction is observable for macroscopic objects.

1 / 10