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Quantum Mechanics Foundations

A free GRE Physics quantum mechanics note on wavefunctions and probability, infinite square well energies, the Heisenberg uncertainty principle, and photon and de Broglie wavelength relations.

Concise answer

Quantum states are described by wavefunctions whose squared magnitude is a probability density; confinement quantizes energy (the infinite well's En=n2E1E_n = n^2E_1 ladder), the uncertainty principle bounds Δx Δp\Delta x\,\Delta p, and the photon and de Broglie relations tie energy and momentum to frequency and wavelength.

Definitions

Wavefunction
The complex function $\Psi(x, t)$ describing a quantum state; $|\Psi|^2$ is the probability density for position measurements.
Normalization
The requirement that the total probability of finding the particle somewhere equals one: $\int |\Psi|^2\,dx = 1$.
Infinite square well
An idealized box with impenetrable walls; the confined particle's allowed energies are discrete and scale as $n^2$.
Uncertainty principle
The statement that position and momentum uncertainties obey $\Delta x\,\Delta p \ge \hbar/2$; sharpening one broadens the other.
De Broglie wavelength
The wavelength $\lambda = h/p$ associated with any particle of momentum $p$, which sets the scale of matter-wave interference.

Intuition

A confined wave must fit whole half-wavelengths between the walls, exactly like a vibrating string. Fewer, longer humps mean lower momentum and energy — that fitting condition is where energy quantization comes from.

Squeezing a particle into a smaller region (Δx\Delta x down) forces a larger momentum spread (Δp\Delta p up). That is why bound quantum systems have a nonzero ground-state energy: perfect stillness at a known location would violate the uncertainty bound.

Concept walkthrough

The wavefunction carries all the information about a quantum state, but only ∣Ψ∣2|\Psi|^2 is measurable — it is the probability density, and integrating it over a region gives the probability of finding the particle there. Normalization fixes the overall amplitude so total probability is one.

The infinite square well is the exam's favorite bound system. Fitting standing waves into a box of width LL gives En=n2h2/(8mL2)E_n = n^2h^2/(8mL^2): levels rise as n2n^2, spacing grows with nn, energy falls as the box widens or the mass increases, and n=1n = 1 has nonzero energy. A photon emitted in a transition carries exactly the level difference ΔE=hf\Delta E = hf.

The uncertainty principle Δx Δp≥ℏ/2\Delta x\,\Delta p \ge \hbar/2 is a property of waves, not a statement about clumsy instruments. Together with the photon relations E=hf=hc/λE = hf = hc/\lambda and the de Broglie relation λ=h/p\lambda = h/p, it links the particle and wave pictures: light of wavelength λ\lambda carries momentum h/λh/\lambda, and electrons of momentum pp diffract like waves of wavelength h/ph/p.

After this page, you should be able to

  • Interpret $|\Psi|^2$ as a probability density and use normalization to fix a wavefunction's amplitude.
  • Compute infinite square well energy levels and photon energies for transitions between them.
  • Apply the Heisenberg uncertainty principle to estimate minimum momentum spreads and ground-state energies.
  • Convert between photon energy, frequency, and wavelength, and compute de Broglie wavelengths for particles.

Formulas and assumptions

Born probability rule

P(a≤x≤b)=∫ab∣Ψ(x)∣2 dxP(a \le x \le b) = \int_a^b |\Psi(x)|^2\,dx

Variables

  • Psi: wavefunction (units of one over square root of meters in one dimension)
  • P: dimensionless probability of finding the particle in [a, b]

Assumptions

  • The wavefunction is normalized so the integral over all space equals one.

Infinite square well energies

En=n2h28mL2E_n = \dfrac{n^2 h^2}{8mL^2}

Variables

  • n: quantum number 1, 2, 3, ...
  • h: Planck's constant in joule-seconds
  • m: particle mass in kilograms
  • L: well width in meters
  • E_n: level energy in joules

Assumptions

  • The walls are infinitely high, so the wavefunction vanishes at both walls.
  • Levels scale as n^2 from the ground-state energy E_1.

Heisenberg uncertainty principle

Δx Δp≥ℏ2\Delta x\,\Delta p \ge \dfrac{\hbar}{2}

Variables

  • Delta x: position uncertainty in meters
  • Delta p: momentum uncertainty in kilogram-meters per second
  • hbar: reduced Planck constant h / (2 pi)

Assumptions

  • The uncertainties are standard deviations of repeated measurements on identically prepared states.
  • The bound is a fundamental wave property, not an instrument limitation.

Photon energy

E=hf=hcλE = hf = \dfrac{hc}{\lambda}

Variables

  • E: photon energy in joules
  • f: frequency in hertz
  • lambda: wavelength in meters
  • c: speed of light in meters per second

Assumptions

  • Light exchanges energy in whole photons.
  • A handy shortcut: hc is approximately 1240 electron-volt nanometers.

De Broglie wavelength

λ=hp\lambda = \dfrac{h}{p}

Variables

  • lambda: matter wavelength in meters
  • h: Planck's constant in joule-seconds
  • p: particle momentum in kilogram-meters per second

Assumptions

  • For nonrelativistic particles p = mv; use relativistic momentum at speeds near c.

Worked example

Electron in a nanometer-wide well

An electron (m=9.11×10−31 kgm = 9.11\times 10^{-31}\,\text{kg}) is confined to an infinite square well of width L=1.0 nmL = 1.0\,\text{nm}. Find the ground-state energy in electron-volts and the energy of the photon emitted in the n=2→n=1n = 2 \to n = 1 transition.

  1. 1Ground state: E1=h28mL2E_1 = \dfrac{h^2}{8mL^2} with h=6.626×10−34 J⋅sh = 6.626\times 10^{-34}\,\text{J}\cdot\text{s}. Numerator: h2=(6.626×10−34)2≈4.39×10−67 J2⋅s2h^2 = (6.626\times 10^{-34})^2 \approx 4.39\times 10^{-67}\,\text{J}^2\cdot\text{s}^2.
  2. 2Denominator: 8mL2=8(9.11×10−31)(1.0×10−9)2≈7.29×10−48 kg⋅m28mL^2 = 8(9.11\times 10^{-31})(1.0\times 10^{-9})^2 \approx 7.29\times 10^{-48}\,\text{kg}\cdot\text{m}^2.
  3. 3Divide: E1≈4.39×10−67/7.29×10−48≈6.0×10−20 JE_1 \approx 4.39\times 10^{-67}/7.29\times 10^{-48} \approx 6.0\times 10^{-20}\,\text{J}. Convert: E1≈6.0×10−20/1.602×10−19≈0.38 eVE_1 \approx 6.0\times 10^{-20}/1.602\times 10^{-19} \approx 0.38\,\text{eV}.
  4. 4Scale by n2n^2: E2=4E1≈1.5 eVE_2 = 4E_1 \approx 1.5\,\text{eV}, so the emitted photon carries ΔE=E2−E1=3E1≈1.1 eV\Delta E = E_2 - E_1 = 3E_1 \approx 1.1\,\text{eV}.

E1≈0.38 eVE_1 \approx 0.38\,\text{eV}, and the n=2→n=1n = 2 \to n = 1 photon carries ΔE=3E1≈1.1 eV\Delta E = 3E_1 \approx 1.1\,\text{eV}.

Common traps

  • Treating Ψ\Psi itself as the probability density; only ∣Ψ∣2|\Psi|^2 is observable, and Ψ\Psi can be negative or complex.
  • Starting the infinite well ladder at n=0n = 0; the lowest state is n=1n = 1 and its energy is nonzero.
  • Forgetting that well energies scale as 1/L21/L^2: doubling the width cuts every level by a factor of four, not two.
  • Using λ=h/p\lambda = h/p with the wrong momentum — for electrons accelerated through a potential difference, get pp from the kinetic energy first.
  • Reading the uncertainty principle as measurement clumsiness; it bounds the state itself, so no better instrument can beat it.

Related pages and practice

GRE Physics overviewSimple Harmonic MotionInterference and DiffractionSpecial Relativity EssentialsSign in for current practice

Question depth and domain coverage vary by exam. Practice answers are checked after submission.

Sources

  1. GRE Subject Test Content and Structure — ETS. Accessed 2026-07-06. Use as a cited source for exam facts; do not imply affiliation or reproduce protected test material.
  2. University Physics Volume 3, Section 7.1: Wave Functions — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
  3. University Physics Volume 3, Section 7.2: The Heisenberg Uncertainty Principle — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
  4. University Physics Volume 3, Section 7.4: The Quantum Particle in a Box — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
  5. University Physics Volume 3, Section 6.2: Photoelectric Effect — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
  6. University Physics Volume 3, Section 6.5: De Broglie's Matter Waves — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.

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