Quantum Mechanics 101 · Unit 4: Atoms, Spin & Identity

Quantum Academy · Chapter 16: The Hydrogen Atom & Quantum Numbers

Orbitals, not orbits: the real shape of an atom

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Instructions. Work through every question with the lesson closed, then check yourself. Use these facts: Eₙ = −13.6 eV/n²; l runs from 0 to n − 1 and mₗ from −l to +l; mₛ = ±½; an orbital has l angular and n − l − 1 radial nodes; a₀ = 0.0529 nm; the probability of finding a 1s electron within distance R of the nucleus is 1 minus the product of (1 + 2x + 2x²) and e to the power −2x, with x = R/a₀; the Bohr magneton μ_B = 9.274×10⁻²⁴ J/T = 5.79×10⁻⁵ eV/T; ħ = 1.055×10⁻³⁴ J·s; h = 6.626×10⁻³⁴ J·s. Show every step in the calculation questions. One point for each vocabulary match and each question; the challenge is a bonus.

Part A Vocabulary matching

Write the letter of the matching definition in the blank beside each term.

1.Orbital
2.Principal quantum number
3.Orbital quantum number
4.Magnetic quantum number
5.Spin quantum number
6.Subshell
7.Node
8.Radial probability
9.Bohr radius
10.Degeneracy
11.Fine structure
12.Zeeman effect

A.mₛ = +½ or −½; labels the component of the electron's intrinsic angular momentum, +ħ/2 or −ħ/2, along a chosen axis.
B.All the orbitals that share the same n and l, such as the three 2p orbitals; it contains 2l + 1 orbitals.
C.mₗ = −l to +l; sets the orbital's orientation and the z-component of its angular momentum, mₗħ, and so its shift in a magnetic field.
D.A surface where the wavefunction, and so the probability density, is zero: angular nodes are planes or cones through the nucleus, radial nodes are spheres.
E.The probability per unit distance of finding the electron at distance r from the nucleus, 4πr² times |ψ|² for an s orbital; it peaks at a₀ for 1s.
F.The splitting of spectral lines by a magnetic field, which separates states with different mₗ and, through its anomalous form, revealed electron spin.
G.The sharing of one energy by several distinct states; hydrogen's level n holds n² spatial states, 2n² with spin.
H.n = 1, 2, 3 …; sets the energy, −13.6 eV/n² in hydrogen, the overall size of the cloud and the total number of nodes, n − 1.
I.The small splitting of hydrogen's levels, roughly α² ≈ 5×10⁻⁵ of their energy, caused by relativity and spin-orbit coupling.
J.A single-electron wavefunction ψₙₗₘ, a stationary standing-wave state whose squared magnitude gives the probability density; not a path and not a container.
K.l = 0 to n − 1; sets the orbital's shape (s, p, d, f) and the magnitude of its orbital angular momentum, √(l(l + 1))·ħ.
L.a₀ = 4πε₀ħ²/(mₑe²) ≈ 0.0529 nm, the most probable distance of hydrogen's 1s electron from the nucleus.

Part B Questions

1.
Which set of quantum numbers (n, l, mₗ, mₛ) describes an allowed state of hydrogen's electron?Circle one
1. A(2, 2, 0, +½)
2. B(3, 1, −2, −½)
3. C(4, 3, −3, +½)
4. D(1, 0, 1, −½)
2.
How many radial and angular nodes does a 4d orbital have?Circle one
1. A3 radial, 0 angular
2. B1 radial, 2 angular
3. C2 radial, 2 angular
4. D0 radial, 3 angular
3.
Ignoring fine structure, how do the energies of hydrogen's 3s, 3p and 3d orbitals compare?Circle one
1. A3s is lowest, then 3p, then 3d
2. B3d is lowest, then 3p, then 3s
3. CAll three are equal, at −1.51 eV
4. DThey depend on the orientation, mₗ
4.
The probability density |ψ|² of hydrogen's 1s state is greatest at a distance a₀ from the nucleus.Circle one
TrueFalse
5.
For a 2p orbital pointing along the z axis, the probability density is zero everywhere on the plane through the nucleus perpendicular to z.Circle one
TrueFalse
6.
Hydrogen's ground-state electron has orbital angular momentum ħ, as Bohr proposed in 1913.Circle one
TrueFalse
7.
The quantum number mₗ is called the quantum number, because a field of that kind splits states with different mₗ.Fill in the blank
8.
A surface on which an orbital's wavefunction is exactly zero is called a .Fill in the blank
9.
Using the idea of a stationary state, explain why it is wrong to picture the 1s electron circling the nucleus.Short answer
10.
A classmate says: 'An orbital is a bubble that the electron lives inside.' Write two or three sentences correcting this.Short answer
11.
For hydrogen's n = 5 level, list the subshells, count the spatial states with and without spin, and find the energy.Show your working
Working
Answer:
12.
Find the probability that a 1s electron is found within 0.5a₀ of the nucleus, and the probability that it is found farther than 3a₀ away.Show your working
Working
Answer:
13.
A hydrogen atom in a 3d state sits in a 1.50 tesla magnetic field. Ignoring spin, into how many sublevels does the 3d level split, what is the energy spacing between neighbouring sublevels in eV, what is the total spread from the lowest to the highest, and what frequency corresponds to one spacing?Show your working
Working
Answer:
14.
For an electron in a d orbital, find the magnitude of the orbital angular momentum in J·s, its largest possible z-component, and the smallest possible angle between the angular momentum and the z axis.Show your working
Working
Answer:

Part C Mad scientist challenge bonus

The 1s electron meets the proton. The probability of finding a 1s electron within distance R of the nucleus is 1 minus the product of (1 + 2x + 2x²) and e to the power −2x, with x = R/a₀. (a) Check the formula: show that its rate of change with x equals 4x² times e to the power −2x, the 1s radial probability written in terms of x, and that it tends to 1 for large R. (b) Show that for very small x the probability is approximately (4/3)x³, and derive the same result directly from the density at the nucleus, |ψ(0)|² = 1/(πa₀³). (c) Taking the proton's radius as 0.84 fm, estimate the probability that the electron is found inside the proton. (d) Explain why the corresponding probability for a 2p electron is far smaller still, and why the tiny overlap of s electrons with the nucleus matters in real physics.

Working

Memory aid

Mnemonic The Four S's, then Walls and Balls. Name each quantum number by what it sets: n is Size, and with it the energy; l is Shape; mₗ is Slant, the orientation; mₛ is Spin. Chant the limits: l stays under n, m sweeps from minus l to plus l, spin flips two ways. Then the node rhyme: 'n minus one in all; l are walls, the rest are balls.' Walls are the angular nodes, flat planes or cones cutting through the nucleus; balls are the spherical radial nodes. To use it on any label such as 4d: Size n = 4, so −13.6 ÷ 16 = −0.85 eV; Shape d, so l = 2; Slant gives five orientations; Spin doubles them to ten states; nodes: three in all, two walls and one ball.

In one line Hydrogen's electron is a 3D standing wave labelled n, l, mₗ, mₛ: energy −13.6 eV/n², shapes set by l; a cloud, never an orbit.