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Topic 4 of 12

Bound States

Bound states are the quantum solutions confined by a potential — from simple square wells to the hydrogen atom.

Potentials

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Infinite Square Well

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Finite Square Well

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Boxes

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Spherical Well

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Hydrogen Atom

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Central Potential

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Energy levels

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Quantum Tunneling

\phi_E(x) = \begin{cases} Ae^{ik_1x} + Be^{-ik_1x} , & x < -a \\ Ce^{ik_1x} + De^{-ik_1x} , & -a < x < a \\ Fe^{ik_1x} + Ge^{-ik_1x} , & x > a \end{cases}

Josephson Junction

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Quantum Dots

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Practice

Exercises

14 problems

1 of 14

An electron (m = 9.11×10⁻³¹ kg) is trapped in an infinite square well of width L = 1.0 nm. Observe the n=1 standing wave and find the ground state energy E₁ in eV.

The animation shows the n=1 standing wave (blue) and its probability density |ψ₁|² (green dashes) inside an infinite square well of width L = 1 nm. The yellow dashed line marks the energy E₁. Enter E₁ in eV.

E₁ = eV

2 of 14

Using the energy level diagram for the same well (L = 1.0 nm), find E₃ in eV. Note that Eₙ = n²E₁.

The energy level diagram shows Eₙ = n²E₁. The green level (n=3) is highlighted. Use Eₙ = n²E₁ to find E₃ in eV. (A photon animates the n=3→n=1 emission.)

E₃ = eV

3 of 14

Find the energy of the photon emitted in the n=3 → n=1 transition for the electron in the 1.0 nm well, in eV.

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4 of 14

For ψ₁(x) = √(2/L)sin(πx/L) with L=2.0 nm, find the probability P(0 ≤ x ≤ L/2).

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5 of 14

A proton (m=1.67×10⁻²⁷ kg) is in an infinite well of L=0.01 nm. Find E₁ in MeV.

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6 of 14

For an electron in L=1.0 nm, how many states have energy below 10 eV?

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7 of 14

For ψ₂(x) = √(2/L)sin(2πx/L) with L=1.0 nm, find the position of probability maximum closest to x=0 in nm.

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8 of 14

Click on the n = 2 orbital in the Bohr model diagram below to read off its energy. What is E₂ for hydrogen in eV?

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9 of 14

Use the energy level diagram to find the ionization energy of hydrogen — the energy needed to remove the electron from the n=1 ground state.

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10 of 14

Find the wavelength of the n=4→n=2 (Hβ) Balmer transition in nm.

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11 of 14

How many distinct quantum states (including spin) does the n=3 shell of hydrogen have?

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12 of 14

Find the energy of the photon emitted in the first Lyman transition (n=2→n=1) in eV.

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13 of 14

Find the radius of the n=2 Bohr orbit in nm.

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14 of 14

What is the maximum ℓ quantum number for the n=4 shell?

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Key Takeaways

Confinement leads to discrete, quantized energy levels
The infinite square well has $E_n \propto n^2/L^2$ with a nonzero ground-state (zero-point) energy
Hydrogen energy levels depend only on the principal quantum number n
Quantum tunneling lets a particle leak through a classically forbidden barrier