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Chapter 7: Problem 44

What is the Heisenberg uncertainty principle? What is the Schrödinger equation?

Short Answer

Expert verified
The Heisenberg Uncertainty Principle states the inverse relation between the precision of position and momentum of a particle, stating that one cannot be known without increased uncertainty in the other. The Schrödinger Equation is the main equation in quantum mechanics, determining the wave function of a physical system, and is used to predict how a system behaves over time.

Step by step solution

01

Define the Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle is an inherent limitation in the precision with which certain pairs of physical properties, such as the position, \(x\), and the momentum, \(p\), of a particle can simultaneously be known. This principle states that the more precisely the position of the particle is known, the less precisely its momentum can be known, and vice versa. Mathematically, it can be framed as \( \Delta x . \Delta p \geq \hbar / 2 \), where \( \Delta x \) and \( \Delta p \) are the uncertainties in the position and momentum, and \( \hbar \) is the reduced Planck's constant.
02

Define the Schrödinger Equation

The Schrödinger equation is the fundamental equation of quantum mechanics, and it provides a way to calculate the wave function of a physical system and how it changes dynamically in time. The time-dependent form of this equation for a single non-relativistic particle of mass 'm' in a potential 'V' can be written as \(i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \left[-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r},t)\right]\Psi(\mathbf{r},t)\), where \( \Psi(\mathbf{r},t) \) is the wave function of the system, \( \hbar \) is the reduced Planck's constant, \( \nabla^2 \) is the Laplace operator, and 'V' is the potential.

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