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In this section, we give a brief derivation and discussion of Heisenberg ’s uncertainty principle. The derivation follows closely that given in Von Neumann’s Mathematical Foundations of Quantum Mechanics pp 230-232. The system we deal with is one di-mensional with coordinate X ranging ; + : The uncertainty principle is a direct.
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The uncertainty, so defined is manifestly non-negative. If the uncertainty is zero, the vector inside the norm is zero and therefore: ΔA(Ψ) = 0 → A −(A)I Ψ = 0 → AΨ = (A)Ψ, (1.3) and the last equation confirms that the state is indeed an eigenstate of A (note that(A) is a number).
In words, you would say that “on the quantum scale, position and momentum cannot be simultaneously determined,” and that “Δx represents the uncertainty in position and Δ p represents the uncertainty in momentum.”. That would be the end of the statement of the Heisenberg Uncertainty Principle.
This chapter establishes the full significance of this equation, and provides a foundation for its application in the following chapters. Operators in quantum mechanics
The wavefunction is more complicated than in the case for a free particle and it is given by the solution to the most important equation in Quantum Physics - Schr ̈odinger’s wave equation. This is not a simple equation for the wavefunction, (x, t), which is a function of position, x, and time t.
Uncertainty principle for non-commuting operators For non-commuting Hermitian operators, we can establish a bound on the uncertainty in the expectation values of A ˆ and B ˆ :
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\[\left\langle a^{2}\right\rangle=\left\langle\psi\left|\hat{A}^{2}\right| \psi\right\rangle\tag{11.21}\] For example, let’s consider the state \(|\psi\rangle=|+z\rangle\) and the observable spin-\(z\). We expect the uncertainty here to be zero, because we know exactly what we’ll get if we measure spin-\(z\). Let’s see if it works out ...
