Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Assume that the researchers place an atom in a state with n = 100, l = 2. What is the magnitude of the orbital angular momentum L associated with this state? (a) 2 ; (b) 6 ; (c) 200 ; (d) 10,100 .

Short Answer

Expert verified
The magnitude of the orbital angular momentum is 6, option (b).

Step by step solution

01

Understanding the Orbital Angular Momentum Formula

The orbital angular momentum for a given state in quantum mechanics is given by the formula L=l(l+1), where is the reduced Planck constant and l is the azimuthal (or orbital) quantum number. In this problem, l=2.
02

Applying the Formula

Substitute l=2 into the formula to find the magnitude of L. Thus, we compute L=2(2+1)=23=6.
03

Identifying the Correct Option

Looking at the given choices, we see that option (b), 6, matches our calculated result. Therefore, the magnitude of the orbital angular momentum for this state is 6.

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.

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

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Orbital Angular Momentum
Orbital angular momentum is a crucial concept in quantum mechanics related to the motion of particles in an atom. It is analogous to the angular momentum in classical physics. However, in the quantum realm, it takes on discrete values rather than a continuous range.
In quantum mechanics, the orbital angular momentum of an electron in an atom is quantized and determined by the azimuthal quantum number, denoted as l. The quantum number l can take on integer values from 0 up to n1, where n is the principal quantum number.
  • The formula for orbital angular momentum is L=l(l+1), where is the reduced Planck constant.
  • This formula shows that the magnitude of the orbital angular momentum depends on l, and is directly proportional to l(l+1).
  • Each value of l corresponds to a different shape of the electron's orbital, which affects how it moves around the nucleus.
By plugging in the quantum number l=2, we find that the orbital angular momentum is 6, as indicated in the problem's solution.
Quantum Numbers
Quantum numbers are sets of numerical values that provide important information about the quantum state of a particle. They are like the particle's "addresses" in the quantum world and are essential for understanding the arrangement of electrons in atoms.
Every electron in an atom is described by a unique set of quantum numbers:
  • Principal Quantum Number (n): This number indicates the energy level of the electron and can be any positive integer. In the exercise, n=100.
  • Azimuthal (or Orbital) Quantum Number (l): It specifies the shape of the orbital and can range from 0 to n1. In our scenario, l=2.
  • Magnetic Quantum Number (ml): This indicates the orientation of the orbital around the nucleus, varying from l to +l.
  • Spin Quantum Number (ms): It describes the intrinsic angular momentum (or spin) of the electron, typically +12 or 12.
These quantum numbers collectively help predict and describe the behavior and interaction of electrons within an atom.
Reduced Planck Constant
The reduced Planck constant, denoted by , is a fundamental constant in quantum mechanics that appears frequently in equations dealing with quantized quantities. It is derived from the Planck constant h, divided by 2π:=h2π
This smaller version of the Planck constant is particularly useful in the context of angular momentum and its quantization in systems such as atoms.
  • quantifies the discreteness of quantum states, serving as the "scale" for action in quantum mechanical systems.
  • It allows the representation of quantities like L=l(l+1) succinctly and conveniently.
  • The reduced Planck constant is critical in describing phenomena where wave-like properties are fundamental, such as the wave-particle duality of electrons.
Overall, helps in measuring changes in quantum systems, linking the quantum world with the classical understanding through the concept of quantization.

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

A hydrogen atom in a particular orbital angular momentum state is found to have j quantum numbers 72 and 92 . (a) What is the letter that labels the value of l for the state? (b) If n=5, what is the energy difference between the j=72 and j=92 levels?

a) How many different 5g states does hydrogen have? (b) Which of the states in part (a) has the largest angle between L and the z-axis, and what is that angle? (c) Which of the states in part (a) has the smallest angle between L and the z-axis, and what is that angle?

The normalized radial wave function for the 2p state of the hydrogen atom is R2p = (1/24a5)re$$r/2a. After we average over the angular variables, the radial probability function becomes P$$(r) dr = (R2p$$)2r2 dr. At what value of r is P$$(r) for the 2p state a maximum? Compare your results to the radius of the n = 2 state in the Bohr model.

Make a list of the four quantum numbers n,l,ml , and ms for each of the 10 electrons in the ground state of the neon atom. Do not refer to Table 41.2 or 41.3.

A hydrogen atom undergoes a transition from a 2p state to the 1s ground state. In the absence of a magnetic field, the energy of the photon emitted is 122 nm. The atom is then placed in a strong magnetic field in the z-direction. Ignore spin effects; consider only the interaction of the magnetic field with the atom's orbital magnetic moment. (a) How many different photon wavelengths are observed for the 2p 1s transition? What are the m$$l values for the initial and final states for the transition that leads to each photon wavelength? (b) One observed wavelength is exactly the same with the magnetic field as without. What are the initial and final m$$l values for the transition that produces a photon of this wavelength? (c) One observed wavelength with the field is longer than the wavelength without the field. What are the initial and final m$$l values for the transition that produces a photon of this wavelength? (d) Repeat part (c) for the wavelength that is shorter than the wavelength in the absence of the field.

See all solutions

Recommended explanations on Physics Textbooks

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