Expectation Values of Position & Momentum | Basics of Probability Theory | OPERATORS

What are Operators in Quantum mechanics? What is the Expectation value of Position, Momentum, Kinetic Energy, Total Energy etc? This is what we discuss in today's lecture.

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In quantum mechanics, expectation values and operators are essential concepts for describing particle behavior. The position operator (X) and momentum operator (P) represent measurable quantities. The expectation value of position, ⟨x⟩, is the average position calculated from the probability density function |ψ(x)|^2. Similarly, the expectation value of momentum, ⟨p⟩, is computed using the probability density function and the momentum operator, often expressed as -iħ(d/dx). These operators and their expectation values provide a probabilistic framework for understanding the behavior of particles, with the uncertainty principle highlighting the inherent limitations in simultaneously measuring position and momentum precisely.

In this video I describe basics of probability theory, derive the expectation values of position, momentum, kinetic energy and total energy. I also discuss the operators corresponding to these quantities.

00:00 Introduction
04:15 Basics of Probability Theory
14:35 Expectation Value of Position
26:58 Expectation Value of Momentum
42:52 Expectation Values & Operators
48:38 Kinetic Energy Operator
51:29 Total Energy Operator

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