Chapter 38 - Photons and Matter Waves - Problems - Page 1184: 69

The position of a particle is measured, not to infinite precision, but to within a distance of $\frac{\lambda}{2\pi}$, where $\lambda$ is the particle’s de Broglie wavelength $\Delta x\leq\frac{\lambda}{2\pi}$ or, $\Delta x\leq\frac{h}{2\pi p}$ According to the Heisenberg’s uncertainty principle $\Delta x\Delta p_x\geq\frac{h}{2\pi}$ or, $\Delta p_x\geq\frac{h}{2\pi\Delta x}$ or, $\Delta p_x\geq\frac{h}{2\pi}\times \frac{2\pi p}{h}$ or, $\Delta p_x\geq p$ Thus, the uncertainty in the momentum component is then equal to the component itself; that is, $\Delta p_x=p$ According to the given data, the position $\Delta x$ is fixed. Therefore, according to the Heisenberg’s uncertainty principle, the momentum component $\Delta p_x$ of a particle is completely known, and which is non-zero. Measured momentum of $0.5p$ is given by $\Delta p_x\geq 0.5p$ Measured momentum of $2p$ is given by $\Delta p_x\geq 2p$ Measured momentum of $12p$ is given by $\Delta p_x\geq 12p$

The position of a particle is measured, not to infinite precision, but to within a distance of $\frac{\lambda}{2\pi}$, where $\lambda$ is the particle’s de Broglie wavelength $\Delta x\leq\frac{\lambda}{2\pi}$ or, $\Delta x\leq\frac{h}{2\pi p}$ According to the Heisenberg’s uncertainty principle $\Delta x\Delta p_x\geq\frac{h}{2\pi}$ or, $\Delta p_x\geq\frac{h}{2\pi\Delta x}$ or, $\Delta p_x\geq\frac{h}{2\pi}\times \frac{2\pi p}{h}$ or, $\Delta p_x\geq p$ Thus, the uncertainty in the momentum component is then equal to the component itself; that is, $\Delta p_x=p$ According to the given data, the position $\Delta x$ is fixed. Therefore, according to the Heisenberg’s uncertainty principle, the momentum component $\Delta p_x$ of a particle is completely known, and which is non-zero. Measured momentum of $0.5p$ is given by $\Delta p_x\geq 0.5p$ Measured momentum of $2p$ is given by $\Delta p_x\geq 2p$ Measured momentum of $12p$ is given by $\Delta p_x\geq 12p$