Quantum Mechanics

Particle in a Box

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Parameters

Quantum Number (n)1
Well Width (L)1.0 nm
Particle Mass1.0 mₑ
Show |ψ|²
Superposition
Quantum # (n)
1
Energy
0.0000 eV
λ (de Broglie)
0.000 nm
P(x = L/2)
0.0000
Δx · Δp / ℏ
0.0000 ⚠
Live Energy Equation
E1=12π2ℏ22⋅1.0me⋅(1.0 nm)2=0.0000 eVE_{1} = \frac{1^2 \pi^2 \hbar^2}{2 \cdot 1.0m_e \cdot (1.0\text{ nm})^2} = 0.0000\text{ eV}
Live Momentum
p1=1πℏ1.0 nm=—×10−24 kg⋅m/sp_{1} = \frac{1\pi\hbar}{1.0\text{ nm}} = — \times 10^{-24}\text{ kg·m/s}
Uncertainty Relation
Δx⋅Δp=0.000 ℏ≥12ℏ\Delta x \cdot \Delta p = 0.000\,\hbar \geq \tfrac{1}{2}\hbar

Awaiting Telemetry

Governing Dynamics

A particle of mass m is confined in a 1D box of width L by infinite potential walls. Inside, V=0; outside, V=∞.

−ℏ22md2ψdx2=Eψ-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\psi

Boundary conditions enforce ψ(0) = ψ(L) = 0, yielding quantized solutions:

ψn(x)=2Lsin⁡ ⁣(nπxL)\psi_n(x) = \sqrt{\frac{2}{L}}\sin\!\left(\frac{n\pi x}{L}\right)n=1,2,3,…n = 1, 2, 3, \ldots
En=n2π2ℏ22mL2E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}

The de Broglie wavelength inside the well:

λn=2Ln,pn=nπℏL\lambda_n = \frac{2L}{n}, \quad p_n = \frac{n\pi\hbar}{L}

Orthonormality of eigenstates:

∫0Lψm∗(x) ψn(x) dx=δmn\int_0^L \psi_m^*(x)\,\psi_n(x)\,dx = \delta_{mn}

The uncertainty principle is always satisfied: Δx·Δp ≥ ℏ/2.