Quiz on Hydrogen Atom and Schrödinger Equation

What does the Hamiltonian operator represent in the context of the hydrogen atom?

• The total energy of the system

• The mass of the electron

• The charge of the proton

• The speed of light

Which equation is referred to as the Time-Independent Schrödinger Equation (TISE) for the hydrogen atom?

• F = ma

• ∇²ψ + V(r)ψ = Eψ

• E = mc²

• Ĥψ = Eψ

What is the form of the electrostatic potential energy for the hydrogen atom?

• V(r) = e²/r

• V(r) = -e²/r

• V(r) = kq₁q₂/r

• V(r) = -kq₁q₂/r

In spherical polar coordinates, what is the Laplacian operator represented as?

• ∇² = (1/r)(∂/∂r)(r ∂/∂r) + (1/r²)(∂/∂θ) + (1/r²sinθ)(∂/∂φ)

• ∇² = (1/r)(∂/∂r)(r ∂/∂r) + (1/r²)(∂²/∂θ²) + (1/r²sin²θ)(∂²/∂φ²)

• ∇² = (1/r²)(∂/∂r)(r² ∂/∂r) + (1/r²sinθ)(∂/∂θ)(sinθ ∂/∂θ) + (1/r²sin²θ)(∂²/∂φ²)

• ∇² = (1/r)(∂/∂r)(r ∂/∂r) + (1/r²)(∂²/∂θ²) + (1/r²)(∂²/∂φ²)

What does the term 'central potential' refer to in the context of the hydrogen atom?

• A potential that depends only on the distance from the nucleus

• A potential that varies with time

• A potential that is uniform throughout space

• A potential that depends on the angular position

What is the principal quantum number denoted by in the energy eigenvalue equation?

• l

• n

• k

• m

Which of the following is a solution to the radial Schrödinger equation for the hydrogen atom?

• Bessel functions

• Laguerre polynomials

• Hermite polynomials

• Fourier series

What is the significance of the quantum number 'l' in the hydrogen atom?

• It represents the azimuthal quantum number

• It represents the principal quantum number

• It represents the magnetic quantum number

• It represents the total angular momentum

What does the term 'bound state solution' refer to in quantum mechanics?

• A state with negative energy

• A state that is not quantized

• A state with positive energy

• A state with zero energy

What is the form of the wave function for the hydrogen atom?

• ψ(r, θ, φ) = R(r)Z(θ, φ)

• ψ(r, θ, φ) = R(r)Y(θ, φ)

• ψ(r, θ, φ) = C(r)D(θ, φ)

• ψ(r, θ, φ) = A(r)B(θ, φ)