significance of results concerning measurements of position and momentum from commutation relations.
Added by Nuria W.
Step 1
Let's think step by step. Show more…
Show all steps
Your feedback will help us improve your experience
Amit Srivastava and 88 other Physics 102 Electricity and Magnetism educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
A bedrock topic in quantum mechanics is the uncertainty principle. It is discussed mostly for massive objects in Chapter $4,$ but the idea also applies to light: Increasing certainty in knowledge of photon position implies increasing uncertainty in knowledge of its momentum, and vice versa. A single-slit pattern that is developed (like the double-slit pattern of Section 3.6 ) one photon at a time provides a good example. Depicted in the accompanying figure, the pattern shows that photons emerging from a narrow slit are spread allover; a photon's $x$ -component of momentum can be any value over a broad range and is thus uncertain. On the other hand, the $x$ -coordinate of position of an emerging photon covers a fairly small range, for $w$ is small. Using the single-slit diffraction formula $n \lambda=w \sin \theta,$ show that the range of likely values of $p_{x}$, which is roughly $p \sin \theta$, is inversely proportional to the range $w$ of likely position values. Thus, an inherent wave nature implies that the precisions with which the particle properties of position and momentum can be known are inversely proportional.
Figure $38-13$ shows a case in which the momentum component $p_{x}$ of a particle is fixed so that $\Delta p_{x}=0 ;$ then, from Heisenberg's uncertainty principle $(E q .38-28),$ the position $x$ of the particle is completely unknown.From the same principle it follows that the opposite is also true;t hat is, if the position of a particle is exactly known $(\Delta x=0),$ the uncertainty in its momentum is infinite. Consider an intermediate case, in which the position of a particle is measured, not to infinite precision, but to within a distance of $\lambda / 2 \pi,$ where $\lambda$ is the particle's de Broglie wavelength. Show that the uncertainty in the (simultaneously measured) mo- mentum component is then equal to the component itself; that is, $\Delta p_{x}=p .$ Under these circumstances, would a measured momentum of zero surprise you? What about a measured momentum of 0.5$p ?$ Of 2$p ?$ Of 12$p ?$
Quantum mechanics Show that the expectation values $\langle p x\rangle$ and $\langle x p\rangle$ are related by $$ \langle p x\rangle-\langle x p\rangle=\frac{\hbar}{i} $$ This result is described by saying that $p$ and $x$ do not commute and it is intimately related to the uncertainty principle.
Timothy J.
Recommended Textbooks
University Physics with Modern Physics
Physics: Principles with Applications
Fundamentals of Physics
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD