Why does the probability density |Ψ|² slosh back and forth, but the average energy ⟨E⟩ doesn't change?

The sloshing arises from the interference between the n=1 and n=2 states, which have different time-dependent phase factors. This changes the shape of Ψ, and thus |Ψ|², over time. However, ⟨E⟩ is a weighted average of the fixed energies E₁ and E₂. Since the weights (|c₁|² and |c₂|²) are constant, ⟨E⟩ is also constant, illustrating energy conservation for a system not interacting with its environment.

Is a particle really in two energy states at once? What does superposition mean?

Yes, in quantum mechanics, a particle can exist in a superposition of states. This doesn't mean it's sometimes in n=1 and sometimes in n=2. Rather, it is in a new, combined state described by the sum Ψ. When you measure the energy, you will collapse the system to either E₁ or E₂ with probabilities |c₁|² and |c₂|². Before measurement, the particle has no definite energy from this set; it possesses the property of the superposition state itself.

What are the main limitations of the infinite 1D box model?

The infinite walls are an idealization; real potential wells have finite height, allowing for quantum tunneling. The model is also one-dimensional and ignores other degrees of freedom (like spin) and interactions between particles. Despite these simplifications, it correctly captures core quantum concepts—quantization, wave-particle duality, and superposition—making it an essential pedagogical tool.

Does this have any connection to real-world systems?

Absolutely. While idealized, this model approximates the behavior of electrons confined in nanostructures like quantum dots or in certain organic molecules with conjugated pi-electron systems (e.g., butadiene). In these systems, electrons are effectively confined to a region of space, and their allowed energies are quantized, influencing optical and electrical properties.

Particle in a 1D Box

The Particle in a 1D Box is a foundational quantum mechanical model that illustrates the wave-like nature of matter. This simulator visualizes a crucial non-stationary state: the time-dependent superposition of the ground (n=1) and first excited (n=2) energy eigenstates. The total wavefunction is Ψ(x,t) = c₁ ψ₁(x) e^{-iE₁t/ħ} + c₂ ψ₂(x) e^{-iE₂t/ħ}, where ψₙ(x) = √(2/L) sin(nπx/L) are the spatial eigenfunctions for an infinite potential well of length L, and Eₙ = n²π²ħ²/(2mL²) are the corresponding quantized energies. The simulator plots the time evolution of the probability density, |Ψ(x,t)|², which shows a sloshing motion as the relative phase between the two components changes. It also tracks the expectation value of energy, ⟨E⟩ = |c₁|²E₁ + |c₂|²E₂, which remains constant in time for this closed system, demonstrating energy conservation. Key simplifications include the infinite well walls (perfect confinement, no tunneling), a single spatial dimension, and the omission of relativistic effects. By interacting with this model, students learn to interpret superposition states, observe how interference between eigenstates leads to a dynamic probability density, and verify that while the wavefunction evolves, the energy expectation value for a superposition of stationary states is time-independent—a direct consequence of the Schrödinger equation.

Who it's for: Undergraduate students in introductory quantum mechanics or physical chemistry courses learning about wavefunctions, superposition, and time evolution.

Key terms

Quantum Superposition
Wavefunction
Probability Density
Stationary State
Expectation Value
Schrödinger Equation
Energy Eigenstate
Time Evolution

How it works

Infinite square well: discrete levels and spatial nodes. Interference between two energy eigenstates makes the probability density slosh back and forth — the quantum analog of beats.

Frequently asked questions

Why does the probability density |Ψ|² slosh back and forth, but the average energy ⟨E⟩ doesn't change?: The sloshing arises from the interference between the n=1 and n=2 states, which have different time-dependent phase factors. This changes the shape of Ψ, and thus |Ψ|², over time. However, ⟨E⟩ is a weighted average of the fixed energies E₁ and E₂. Since the weights (|c₁|² and |c₂|²) are constant, ⟨E⟩ is also constant, illustrating energy conservation for a system not interacting with its environment.
Is a particle really in two energy states at once? What does superposition mean?: Yes, in quantum mechanics, a particle can exist in a superposition of states. This doesn't mean it's sometimes in n=1 and sometimes in n=2. Rather, it is in a new, combined state described by the sum Ψ. When you measure the energy, you will collapse the system to either E₁ or E₂ with probabilities |c₁|² and |c₂|². Before measurement, the particle has no definite energy from this set; it possesses the property of the superposition state itself.
What are the main limitations of the infinite 1D box model?: The infinite walls are an idealization; real potential wells have finite height, allowing for quantum tunneling. The model is also one-dimensional and ignores other degrees of freedom (like spin) and interactions between particles. Despite these simplifications, it correctly captures core quantum concepts—quantization, wave-particle duality, and superposition—making it an essential pedagogical tool.
Does this have any connection to real-world systems?: Absolutely. While idealized, this model approximates the behavior of electrons confined in nanostructures like quantum dots or in certain organic molecules with conjugated pi-electron systems (e.g., butadiene). In these systems, electrons are effectively confined to a region of space, and their allowed energies are quantized, influencing optical and electrical properties.

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Launch Simulator

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Rectangular Barrier Tunneling

Analytic transmission T(E) vs energy for a 1D barrier; tunneling below V₀ and resonances above (ℏ = m = 1).

Launch Simulator

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Stern–Gerlach Beam (Cartoon)

Silver-like beam through an inhomogeneous B-field: atoms deflect to two detectors illustrating S_z = ±ℏ/2.

Launch Simulator

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Bloch Sphere & Rabi Drive

Two-level Bloch vector with du/dt = u × ω, ω = (0, Ω, Δ); u_x–u_z view, Rabi flopping when Δ = 0.

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Hydrogen |ψ|² (xz slice)

Coulomb hydrogen Z=1: |R_nl|² × |Y_lm|² colormap in the xz plane for textbook n,l states.

Launch Simulator

Particle in a 1D Box

How it works

Frequently asked questions

More from Chemistry

Gaussian Wave Packet

Quantum Well Eigenstates (Box & HO)

Rectangular Barrier Tunneling

Stern–Gerlach Beam (Cartoon)

Bloch Sphere & Rabi Drive

Hydrogen |ψ|² (xz slice)

Particle in a 1D Box

How it works

Frequently asked questions