Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 5: Problem 19

We have seen that the angular momentum commutation rules are generated by considering consecutive infinitesimal rotations about perpendicular axes in three-dimensional space. Could it be that the fundamental quantum mechanical commutation rule \(\left[x, p_{x}\right]=\mathrm{i} \hbar\) is also just a manifestation of the geometry of three-dimensional space? Present an argument that makes use of the angular momentum result and the definition of the angular momentum operators in terms of position and linear momentum operators that could be used to justify this supposition.

Short Answer

Expert verified
The fundamental quantum mechanical commutation rule \(\left[x, p_{x}\right]=\mathrm{i} \hbar\) could indeed be a manifestation of geometry in three-dimensional space, considering angular momentum commutation relation and the definition of the angular momentum operators in terms of position and linear momentum operators. This is reflected by considering a small displacement along \(x\) and its corresponding momentum change, which give rise to a small angular momentum in three-dimensional space.

Step by step solution

01

Understand the Concept of Commutation and Angular Momentum

Commutation rules of angular momentum are given by \(\left[L_{i}, L_{j}\right]=\mathrm{i} \hbar \varepsilon_{ijk} L_{k}\), where \(i, j, k\) are x, y, z axes and \(L_{k}\) is the angular momentum operator in the \(k\) direction. Recall that the angular momentum operators are defined in terms of position and linear momentum operators as \(L_{k} = \varepsilon_{ijk} x_{i} p_{j}\).
02

Interpret the Linear Momentum Commutation Rule

The quantum mechanical commutation rule for linear momentum is \(\left[x, p_{x}\right]=\mathrm{i} \hbar\). This rule can be seen in terms of angular momentum operators by modifying the definition of \(L_{k}\) slightly and setting \(i =j =x : L_{k} = \varepsilon_{x x k} x_{x} p_{x}\). This simplifies to \(L_{k} = x p_{x}\).
03

Manifest the Geometry in Three-Dimensional Space

For a geometric interpretation, consider a small displacement along \(x\), \(dx\), and its corresponding momentum change \(dp_{x}\). Their cross product in three-dimensional space would give rise to a small angular momentum, \(dL_{k}\), about the \(k\) axis. Hence, the commutation rule of \(x\) and \(p_{x}\) reflects the geometry of the three-dimensional space.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the representatives of the operations of the group \(T_{\mathrm{d}}\) by using as a basis four 1 s-orbitals, one at each apex of a regular tetrahedron (as in \(\mathrm{CH}_{4}\) ). Hint. The basis is four-dimensional; the order of the group is \(24,\) and so there are 24 matrices to find.

Analyse the following direct products into the symmetry species they span: (a) \(C_{2 v}: A_{2} \times B_{1} \times B_{2}\) (b) \(C_{3 v}: A_{1} \times A_{2} \times E\) (c) \(C_{6 v}: B_{2} \times E_{1},(d) C_{o v v}: E_{1}^{2}\) (e) \(\mathrm{O}: \mathrm{T}_{1} \times \mathrm{T}_{2} \times \mathrm{E}\)

Classify the terms that may arise from the following configurations: (a) \(C_{2 v}: a_{1}^{2} b_{1}^{1} b_{2}^{1} ;\) (b) \(C_{3 v}: a_{2}^{1} e^{1}, e^{2} ;(c) T_{d}: a_{2}^{1} e^{1}\) \(\mathrm{e}^{1} \mathrm{t}_{1}^{1}, \mathrm{t}_{1}^{1} \mathrm{t}_{2}^{1}, \mathrm{t}_{1}^{2}, \mathrm{t}_{2}^{2} ;\) (d) \(\mathrm{O}: \mathrm{e}^{2}, \mathrm{e}^{1} \mathrm{t}_{1}^{1}, \mathrm{t}_{2}^{2} .\) Hint. Use the direct- product tables; triplet terms have antisymmetric spatial functions.

Show that in an octahedral array, hydrogen 1 s-orbitals \(\operatorname{span} \mathrm{A}_{1 \mathrm{g}}+\mathrm{E}_{\mathrm{g}}+\mathrm{T}_{1 \mathrm{u}}\) of the group \(\mathrm{O}_{\mathrm{h}}\)

Show that \(3 x^{2} y-y^{3}\) is a basis for an \(\mathrm{A}_{1}\) irreducible representation of \(C_{3 v} .\) Hint. Show that \(C_{3}^{+}\left(3 x^{2} y-y^{3}\right) \propto\) \(3 x^{2} y-y^{3} ;\) likewise for the other elements of the group.
See all solutions

Recommended explanations on Chemistry Textbooks

Chemical Analysis

Read Explanation

The Earths Atmosphere

Read Explanation

Ionic and Molecular Compounds

Read Explanation

Nuclear Chemistry

Read Explanation

Organic Chemistry

Read Explanation

Kinetics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free