Spherical Polar Coordinates Relationship Equations Quiz

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Questions and Answers

In the equations provided, what does 𝐱 represent?

The x-coordinate

What type of coordinates should be used based on the given text?

Spherical polar coordinates

Which equation should be substituted into eq. 4 according to the text?

Equation for 𝐳

What is the role of 𝛉 in the equations provided in the text?

Changes the orientation of the coordinates Signup and view all the answers

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

Sch Signup and view all the answers

What does the magnetic quantum number represent?

The separation of the variable in the wave function Signup and view all the answers

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

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

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

It simplifies the equation and solves for 𝛉 more efficiently Signup and view all the answers

What does the solution of the 𝜽 equation depend on?

The separation of variables in the wave function Signup and view all the answers

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

It simplifies the solution process for 𝛉 Signup and view all the answers

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

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

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

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

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

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

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

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

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

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

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

Real-valued Signup and view all the answers

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

Maximum is +1, minimum is -1 Signup and view all the answers

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

The orbital angular momentum quantum number Signup and view all the answers

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

To simplify the radial equation for the hydrogen atom Signup and view all the answers

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

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

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

To introduce a dimensionless radial coordinate 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?

The total energy must be negative Signup and view all the answers

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

The radial distance Signup and view all the answers

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

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

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

$\rho = \infty$ Signup and view all the answers

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

It represents the separation constant Signup and view all the answers

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

The equation is a second-order partial differential equation Signup and view all the answers

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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).