Question

The Heisenberg uncertainty principle can be expressed in the form $$ \Delta E \cdot \Delta t \geqq \frac{h}{4 \pi} $$ where \(E\) represents energy and \(t\) represents time. Show that the units for this form are the same as the units for the form used in this chapter: $$ \Delta x \cdot \Delta(m v) \geq \frac{h}{4 \pi} $$

Short answer

In both forms of the Heisenberg uncertainty principle, the units are the same: Joule-seconds (Js). For the first form \(\Delta E \cdot \Delta t = J \cdot s\), and for the second form \(\Delta x \cdot \Delta(mv) = J \cdot s\). Also, the units of the Planck constant (\(h\)) term are \(J \cdot s\). Thus, both forms have equivalent units, validating the Heisenberg uncertainty principle for both expressions.

Step-by-step solution

Units of Terms in the First Form

For the first form, we need to find the units of \(\Delta E \cdot \Delta t\). The units of energy (\(E\)) are Joules (J) and the units of time (\(t\)) are seconds (s). So, \(\Delta E \cdot \Delta t = J \cdot s\) Now let's find the units of the terms in the second form.

Units of Terms in the Second Form

For the second form, we need to find the units of \(\Delta x \cdot \Delta(mv)\). The position \(x\) has units of meters (m), and the product of mass (\(m\)) and velocity (\(v\)) has units of kg m/s. Therefore, \(\Delta x \cdot \Delta(mv) = m \cdot (kg \cdot m/s) = J \cdot s\)

Units of the Constant `h`

The constant `h` is the Planck constant, and it has units of Joule-seconds (Js). The term \(\frac{h}{4\pi}\) in both forms has units of \(J\cdot s\): \(\frac{h}{4\pi} = \frac{Js}{4\pi}\)

Comparing the Units

Now, we compare the units of both forms: 1. First form: \(\Delta E \cdot \Delta t \geqq \frac{h}{4 \pi}\) has units of \(J \cdot s \geqq J\cdot s\) 2. Second form: \(\Delta x \cdot \Delta(m v) \geq \frac{h}{4 \pi}\) has units of \(J \cdot s \geq J\cdot s\) Since the units of both forms are the same, the Heisenberg uncertainty principle holds true for both formulations.

Key concepts

Quantum Mechanics

At the heart of the most perplexing phenomena in the universe lies the field of quantum mechanics, a cornerstone of modern physics.

Quantum mechanics delves into the behavior and interactions of subatomic particles, such as electrons and photons, at the smallest scales imaginable. Unlike classical physics, which can predict the motion of planets and projectiles with great precision, quantum mechanics reveals a world where probabilities and uncertainties dominate.

• In this realm, particles can exist in multiple states simultaneously, known as superposition.
• They can affect each other instantly over vast distances—a phenomenon called entanglement.
• Observing a particle can fundamentally alter its state, a concept splashed into the mainstream with Schrödinger's cat thought experiment.

These quantum behaviors often seem counterintuitive because they do not align with our everyday experiences. It's these principles that inform the Heisenberg uncertainty principle, which sets fundamental limits on the precision with which certain pairs of physical properties, like position and momentum, can be known simultaneously.

Planck Constant

The Planck constant, symbolized as `h`, is of paramount importance in quantum mechanics, holding the universe's smallest actions in its dimensional grip.

It was Max Planck who introduced this constant in 1900, proposing that energy is quantized, meaning it could be released or absorbed only in discrete amounts or 'quanta'. This constant is central to the quantum theory and is a key element in several fundamental equations, including the energy of a photon `E = hν`, where `ν` is the frequency of the photon.

• The value of the Planck constant is approximately 6.62607015 × 10^-34 joule-seconds (Js).
• In addition to connecting energy and frequency, `h` also relates to the wavelength of particles via the de Broglie equation `λ = h/p`, with `λ` being the wavelength and `p` the momentum.

Understanding the Planck constant is crucial when dealing with the Heisenberg uncertainty principle, as it sets the scale for the 'quantum of action' that underlies the inherent limitations on the precision of measurements at quantum levels.

Dimensional Analysis

Dimensional analysis is a mathematical tool often used in physics to check the consistency of equations, convert units, or to derive relationships between physical quantities.

It's based on the principle that you can only equate or compare quantities if they have the same dimensions. For instance, adding length to time directly would be like adding apples and oranges – it’s nonsensical from a dimensional perspective.

• One common use is to ensure that the units on both sides of an equation match, affirming that an equation is dimensionally consistent.
• Another application helps in estimating the results of complex calculations just by analyzing the units and dimensions involved.

The solution to the Heisenberg uncertainty principle exercise employs dimensional analysis to verify that both forms of the principle have units that are consistent, thereby showing that they are dimensionally equivalent. The comparison of units, Joule-seconds on both sides, confirms the validity of the expressions. This clear and simple method is a powerful test of physical equations and principles like Heisenberg's, ensuring their dimensional integrity.