Spherical Polar Coordinates Relationship Equations Quiz
27 Questions
0 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

In the equations provided, what does 𝐱 represent?

  • The x-coordinate (correct)
  • The z-coordinate
  • The spherical coordinate
  • The y-coordinate
  • What type of coordinates should be used based on the given text?

  • Spherical polar coordinates (correct)
  • Polar coordinates
  • Rectangular coordinates
  • Cylindrical coordinates
  • Which equation should be substituted into eq. 4 according to the text?

  • Equation for 𝐳 (correct)
  • Equation for 𝐲
  • Equation for 𝐱
  • Equation for 𝐫
  • What is the role of 𝛉 in the equations provided in the text?

    <p>Changes the orientation of the coordinates</p> Signup and view all the answers

    What is obtained by substituting from eq. 5 into eq. 4 according to the text?

    <p>Sch</p> Signup and view all the answers

    What does the magnetic quantum number represent?

    <p>The separation of the variable in the wave function</p> Signup and view all the answers

    In the context of the equation provided, what does the variable z represent?

    <p>Function p(z) as a replacement for variable 𝜽</p> Signup and view all the answers

    What is the significance of substituting p(z) in the equation?

    <p>It simplifies the equation and solves for 𝛉 more efficiently</p> Signup and view all the answers

    What does the solution of the 𝜽 equation depend on?

    <p>The separation of variables in the wave function</p> Signup and view all the answers

    How does introducing a new variable z affect solving the given equation?

    <p>It simplifies the solution process for 𝛉</p> Signup and view all the answers

    What is the equation that is the same as the free particle equation?

    <p>$\frac{\partial^2 \Phi}{\partial \phi^2} = -m^2 \Phi$</p> Signup and view all the answers

    What is the solution to the $\Phi$ equation?

    <p>$\Phi(\phi) = A e^{im\phi}$</p> Signup and view all the answers

    What is the value of the normalization constant $A$?

    <p>$A = \frac{1}{\sqrt{2\pi}}$</p> Signup and view all the answers

    What are the conditions for $\Phi$ to be a single-valued function?

    <p>$\Phi(\phi) = \Phi(\phi + 2\pi)$</p> Signup and view all the answers

    What is the condition on the parameter $m$ for $\Phi$ to be a single-valued function?

    <p>$m = 0, \pm 1, \pm 2, \dots$</p> Signup and view all the answers

    Which of the following is not a property of the $\Phi$ function?

    <p>Real-valued</p> Signup and view all the answers

    What is the maximum and minimum value of the $\Phi$ function?

    <p>Maximum is +1, minimum is -1</p> Signup and view all the answers

    What does the symbol $\ell$ represent in the context of the given text?

    <p>The orbital angular momentum quantum number</p> Signup and view all the answers

    What is the purpose of the substitution $\alpha^2 = -2mE/\hbar^2$?

    <p>To simplify the radial equation for the hydrogen atom</p> Signup and view all the answers

    What is the significance of the term $\beta$ in the given equations?

    <p>It is a term that depends on the orbital angular momentum quantum number $\ell$</p> Signup and view all the answers

    What is the purpose of the substitution $\rho = 2\alpha r$?

    <p>To introduce a dimensionless radial coordinate</p> Signup and view all the answers

    According to the given text, what is the condition for the total energy of an electron in a hydrogen atom?

    <p>The total energy must be negative</p> Signup and view all the answers

    What is the variable used to represent in the given equations?

    <p>The radial distance</p> Signup and view all the answers

    Which of the following is the correct way to express the equation given in the text?

    <p>$\frac{\partial^2 R}{\partial \rho^2} + \frac{1}{\rho} \frac{\partial R}{\partial \rho} + \lambda R = 0$</p> Signup and view all the answers

    What is the value of at which the equation in the text is evaluated?

    <p>$\rho = \infty$</p> Signup and view all the answers

    What is the purpose of the variable in the equation provided?

    <p>It represents the separation constant</p> Signup and view all the answers

    Which of the following can be inferred from the information provided in the text?

    <p>The equation is a second-order partial differential equation</p> Signup and view all the answers

    Study Notes

    Quantum Numbers

    • The physical meaning of quantum numbers: They are used to describe the energy, shape, and orientation of an electron's orbit around the nucleus.

    Magnetic Quantum Number (m)

    • m is called the magnetic quantum number.
    • It describes the orientation of an electron's orbit around the nucleus.

    Orbital Angular Momentum Quantum Number (l)

    • l is called the orbital angular momentum quantum number.
    • It describes the shape of an electron's orbit around the nucleus.

    Solution of θ Equation

    • The equation of θ variable can be separated using a new variable (z), where z = cosθ.
    • The function Φ(θ) will be replaced by p(z), where p(z) = Φ(θ) / sinθ.

    Solution of R Equation

    • The equation of R variable can be separated using spherical polar coordinates.
    • The relation between (x, y, z) and (r, θ, φ) is given by x = r sinθ cosφ, y = r sinθ sinφ, and z = r cosθ.

    Solution of φ Equation

    • The equation of φ variable is the same as the free particle equation.
    • Its solution can be written as: Φ(φ) = Aeimφ.
    • The normalization condition is: ∫|Φ(φ)|² dφ = 1.
    • Φ(φ) satisfies the first condition (finite) and the second condition (single-valued).
    • The maximum value of Φ(φ) is (+1) and the minimum value is (-1).
    • Φ(φ) satisfies the condition: Φ(φ) = Φ(φ + 2π).

    Studying That Suits You

    Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

    Quiz Team

    Description

    Test your understanding of the relationships between Cartesian coordinates (x, y, z) and spherical polar coordinates (r, θ, φ) with this quiz. Practice substituting equations to derive new expressions in spherical polar coordinates.

    More Like This

    Spherical Mastery
    9 questions
    Spherical Polar Coordinates Quiz
    5 questions
    Spherical Mirrors and Image Formation
    28 questions

    Spherical Mirrors and Image Formation

    SpellbindingBananaTree9738 avatar
    SpellbindingBananaTree9738
    Use Quizgecko on...
    Browser
    Browser