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Ch 39: Wave Functions and Uncertainty
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 39: Wave Functions and UncertaintyProblem 43a
Chapter 39, Problem 43a

Heavy nuclei often undergo alpha decay in which they emit an alpha particle (i.e., a helium nucleus). Alpha particles are so tightly bound together that it’s reasonable to think of an alpha particle as a single unit within the nucleus from which it is emitted. A 238U nucleus, which decays by alpha emission, is 15 fm in diameter. Model an alpha particle within a 238U nucleus as being in a one-dimensional box. What is the maximum speed an alpha particle is likely to have?

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Model the alpha particle as a particle in a one-dimensional box. The energy levels of a particle in a one-dimensional box are given by the formula: En = n2h²/8mL2, where n is the quantum number, h is Planck's constant, m is the mass of the particle, and L is the length of the box.
The diameter of the uranium nucleus is given as 15 fm (femtometers), so the length of the one-dimensional box is approximately L = 15 fm = 15 × 10-15 m.
The mass of the alpha particle is approximately m = 6.64 × 10-27 kg. Use this value for the mass in the energy formula.
The maximum speed corresponds to the maximum energy, which occurs at the first energy level (n = 1). Calculate the energy using the formula for E1 and substitute the values for h, m, and L.
Once the energy is calculated, use the relationship between kinetic energy and speed: E = 12mv2. Solve for v (speed) by rearranging the equation: v = 2E. Substitute the calculated energy and the mass of the alpha particle to find the maximum speed.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quantum Mechanics and Particle in a Box Model

In quantum mechanics, the particle in a box model describes a quantum particle confined to a one-dimensional space with infinitely high potential walls. This model helps to determine the energy levels and wave functions of the particle. For an alpha particle in a nucleus, this model allows us to calculate its maximum speed based on its quantized energy states.
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Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle states that it is impossible to simultaneously know both the position and momentum of a particle with absolute precision. This principle is crucial in quantum mechanics, as it implies that the more precisely we know the position of the alpha particle within the nucleus, the less precisely we can know its momentum, affecting its calculated speed.
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Kinetic Energy and Speed Relationship

The kinetic energy of a particle is directly related to its speed, given by the equation KE = 1/2 mv², where m is the mass and v is the speed. In the context of the alpha particle, understanding this relationship allows us to derive its maximum speed from the energy levels determined by the particle in a box model, providing insight into its behavior during alpha decay.
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Related Practice
Textbook Question

A pulse of light is created by the superposition of many waves that span the frequency range f₀ − (1/2) Δf ≤ f ≤ f₀ + (1/2) Δf, where f₀ = c/λ is called the center frequency of the pulse. Laser technology can generate a pulse of light that has a wavelength of 600 nm and lasts a mere 6.0 fs (1 fs = 1 femtosecond =10−15 s). What is the spatial length of the laser pulse as it travels through space?

Textbook Question

A particle is described by the wave function ψ(x)={cex/Lx≤0 mmce−x/Lx≥0 mm\(\psi\) (x)=\(\begin{cases}\) ce^{x/L} & x\(\leq\) 0\(\text{ mm}\) \\ ce^{-x/L} & x\(\geq\) 0\(\text{ mm}\) \(\end{cases}\) where L = 2.0 mm. Interpret your answer to part b by shading the region representing this probability on the appropriate graph in part a.

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Textbook Question

Heavy nuclei often undergo alpha decay in which they emit an alpha particle (i.e., a helium nucleus). Alpha particles are so tightly bound together that it’s reasonable to think of an alpha particle as a single unit within the nucleus from which it is emitted. The probability that a nucleus will undergo alpha decay is proportional to the frequency with which the alpha particle reflects from the walls of the nucleus. What is that frequency (reflections/s) for a maximum-speed alpha particle within a 238U nucleus?

Textbook Question

Consider the electron wave function ψ(x)={c1−x2∣x∣≤1 cm0∣x∣≥1 cm\(\psi\) (x)=\(\begin{cases}\) c\(\sqrt{1-x^{2}\)} & \(\left\)|x\(\right\)|\(\leq\) 1\(\text{ cm}\) \\ 0 & \(\left\)|x\(\right\)|\(\geq\) 1\(\text{ cm}\) \(\end{cases}\) where x is in cm. If 104 electrons are detected, how many will be in the interval 0.00 cm ≤ x ≤ 0.50 cm?