Question

The Heisenberg uncertainty principle can be expressed in the form $$ \Delta E \cdot \Delta t \geq \frac{h}{4 \pi} $$ 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

The units of the first form of the Heisenberg uncertainty principle, \(\Delta E \cdot \Delta t\), are \(J \cdot s\), which can be expressed as \( kg \cdot m^2/s \). The units of the second form, \(\Delta x \cdot \Delta(mv)\), are also \(kg \cdot m^2/s\). Thus, the units in both forms are compatible.

Step-by-step solution

Understand the First Form Basic Unit

The Heisenberg uncertainty principle states that the uncertainty in the energy of a system multiplied by the uncertainty in the time of the system is greater than or equal to a constant value divided by \(4\pi\). Mathematically, it can be expressed as: \[ \Delta E \cdot \Delta t \geq \frac{h}{4 \pi} \] where \(\Delta E\) is the uncertainty in energy (joules, J), \(\Delta t\) is the uncertainty in time (seconds, s), and \(h\) is the Planck's constant (\(6.626 \times 10^{-34}\) Js).

Understand the Second Form Basic Unit

The Heisenberg uncertainty principle can be understood in another form involving uncertainty in position. Mathematically, it can be expressed as: \[ \Delta x \cdot \Delta(mv) \geq \frac{h}{4 \pi} \] where \(\Delta x\) is the uncertainty in position (meters, m), \(m\) is the mass of the particle (kilograms, kg), and \(v\) is the velocity of the particle (meters per second, m/s). Now, let's verify if the units in both expressions are the same. #Step 2: Verify the Units#

Units of the First Form

In the first form of the Heisenberg uncertainty principle, we have: \[ \Delta E \cdot \Delta t \] The units of energy \(\Delta E\) are joules (J), and the units of time \(\Delta t\) are seconds (s). So, the units for the left-hand side of the equation are \(J \cdot s\).

Units of the Second Form

In the second form of the Heisenberg uncertainty principle, we have: \[ \Delta x \cdot \Delta(mv) \] The units of position \(\Delta x\) are meters (m), the units of mass \(m\) are kilograms (kg), and the units of velocity \(v\) are meters per second (m/s). So, the units for the left-hand side of the equation are \((m)\cdot(kg \cdot m/s)\) or \(kg \cdot m^2/s\). Now, let's see if the units in both forms are compatible. #Step 3: Check for Compatibility of Units#

Compatibility Check

In the first form of the Heisenberg uncertainty principle, the units of the left-hand side are \(J \cdot s\). As we know, \(1J = 1kg \cdot m^2/s^2\), so the units in the first form can be rewritten as: \[ kg \cdot m^2/s \] For the second form of the Heisenberg uncertainty principle, the units of the left-hand side are \(kg \cdot m^2/s\). Since the units of the left-hand side in both forms of the Heisenberg uncertainty principle are the same, we can conclude that the units of both forms are compatible.

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It incorporates principles such as the Heisenberg uncertainty principle, which sets inherent limits on how precisely we can simultaneously measure certain pairs of physical properties, like position and momentum. The uncertain nature of quantum particles is not due to the limits of measurement but is an intrinsic quality of quantum systems.

In quantum mechanics, particles are treated as waves with a certain probability of being found in a particular location. This wave-particle duality is crucial for understanding quantum behavior. The equations and principles of quantum mechanics, like those proposed by Heisenberg, Dirac, and Schrödinger, have given rise to numerous technological advances, including semiconductors and lasers, profoundly impacting modern life.

Physical Chemistry

Physical chemistry is the branch of chemistry concerned with the interpretation of the physical properties and compositions of chemical systems using the principles of physics. It involves the study of how matter behaves on a molecular and atomic level and how chemical reactions occur. This field connects the macroscopic world that we can observe with the microscopic world of atoms and molecules.

The Heisenberg uncertainty principle, while stemming from quantum physics, plays a significant role in physical chemistry. It influences our understanding of reaction rates, energy transitions in atoms and molecules, and the behavior of electrons in chemical bonds. By integrating this principle into the study of reaction mechanisms and energetic processes, physical chemists gain insight into how energy and matter interact in chemical systems.

Unit Analysis

Unit analysis, also known as dimensional analysis, is a method used widely in science and engineering to convert one set of units to another and to check the correctness of equations. It's a powerful tool in problem-solving that emphasizes the role of units in calculations, assuring that the final result of a computation has the desired units.

In the context of the Heisenberg uncertainty principle, unit analysis verifies the consistency of different expressions of the principle. As we saw in the step-by-step solution, it confirms that \(\Delta E \cdot \Delta t\) and \(\Delta x \cdot \Delta(mv)\) have compatible units, both amounting to \(kg \cdot m^2/s\). This exercise showcases the importance of unit analysis in validating physical equations and ensuring that they make sense dimensionally, reinforcing students’ understanding of how different physical quantities relate to each other.