Quantum Wave Packet Simulator #88
Real-time 1D Schr�dinger equation solved via split-operator FFT � watch Gaussian packets tunnel, scatter, oscillate, and collide.
Quantum Mechanics Physics
Time-Dependent Schr�dinger Equation
The wave function ?(x,t) encodes all knowable information about a quantum particle.
Its evolution is governed by the time-dependent Schr�dinger equation (TDSE):
ih ∂ψ/∂t = H? = (-h�/2m � ?�/?x� + V(x)) ?
Initial Gaussian wave packet (position-momentum minimum uncertainty state):
?(x,0) = (2ps�)^{-1/4} � exp(-(x-x0)�/4s�) � exp(ik0x)
s = initial position width
k0 = mean wavenumber (initial momentum p0 = hk0)
s_x�s_p = h/2 (minimum uncertainty product)
Free-particle solution (V=0):
?(x,t) = ? f(k) exp(i(kx - ?(k)t)) dk / v(2p)
f(k) = FFT[?(x,0)] (momentum-space representation)
?(k) = hk�/2m (dispersion relation)
Spreading width: s(t) = s0 v(1 + (ht/2ms0�)�)
Group velocity: v_g = d?/dk|_{k=k0} = hk0/m = p0/m
Phase velocity: v_ph = ?/k = hk/2m = v_g/2 (for free particle)
Quantum Tunneling
Rectangular barrier (0 < x < a, V = V0 > E):
Inside barrier: ?(x) = A e^{-?x} + B e^{+?x} ? = v(2m(V0-E))/h
WKB transmission probability:
T � exp(-2?0? ?(x) dx)
Rectangular: T � [1 + (V0�sinh�(?a))/(4E(V0-E))]^{-1}
Thin barrier: T � exp(-2?a) (?a � 1)
Applications:
� alpha decay: Gamow factor G = exp(-2 ?_R^{R_c} ?(r) dr)
� STM tunnel current: I ? exp(-2?d) (d = tip-sample gap)
� tunnel diode: band-to-band tunneling for fast switching
� enzyme catalysis: H-transfer even at zero temperature
Coherent States � Harmonic Oscillator
Harmonic potential: V(x) = �m?�x� E? = (n + �)h?
Energy eigenstates: ψₙ(x) = H?(xv(m?/h)) exp(-m?x�/2h) / normalisation
H? = Hermite polynomial; n = 0,1,2,...
Coherent state |α⟩ (minimum-uncertainty, classical-like):
?_a(x,t) = Gaussian packet that oscillates without spreading
?x?(t) = x_cl�cos(?t + f) (follows classical trajectory)
?p?(t) = -m?x_cl�sin(?t + f)
Width s = v(h/2m?) = constant (no spreading!)
Thermal state (Fock state superposition) shows quantum revivals:
?(x, T_revival) � ?(x, 0) at T_revival = 2p/?
Uncertainty Principle
Heisenberg uncertainty principle:
s_x � s_p = h/2 (Robertson inequality for any state)
s_x � s_p = h/2 (Gaussian: minimum-uncertainty state)
Energy-time uncertainty:
s_E � s_t = h/2 (spectral linewidth ? lifetime)
Narrower packet (smaller s) ? more momentum components in FFT
? faster spreading due to larger ?p
Robertson uncertainty for general operators A, B:
s_A � s_B = �|?[�,B^]?|
For x^, p^: [x^, p^] = ih ? s_x s_p = h/2
Split-Operator FFT Algorithm
Trotter-Suzuki decomposition (second-order):
e^{-iHdt/h} � e^{-iV^dt/2h} � e^{-iT^dt/h} � e^{-iV^dt/2h} + O(dt�)
Split-step algorithm per timestep:
1. ? ? exp(-iV(x)dt/2) � ? [position-space half-step]
2. f ? FFT(?) [to k-space]
3. f ? exp(-ihk�dt/2m) � f [k-space full step]
4. ? ? IFFT(f) [back to x-space]
5. ? ? exp(-iV(x)dt/2) � ? [position-space half-step]
Properties:
� Unitary: exactly preserves ‖ψ‖² = 1 (norm conservation)
� Time-reversible: replace dt ? -dt to run backward
� O(N log N) per step via Cooley-Tukey FFT (N = 512 grid points)
� Global error O(dt�) per step, O(dt�) total (symplectic integrator)
Preset Guide
| Preset | Key Physics | What to observe |
|---|
| 🌊 Free Particle | s(t)=s0v(1+(ht/2ms0�)�) | Packet spreads; reduce s for faster spreading; expectation ?x? moves at constant speed p0/m |
| 〰 Harmonic Osc. | ?x?=A cos(?t); coherent state | Packet oscillates without spreading; width stays constant � unique property of coherent states |
| ⚪ Barrier Tunnel | T�exp(-2?a); ?=v(2m(V-E))/h | Increase V0 ? less transmission; both reflected and transmitted peaks visible as packets |
| ⚛️ Double Well | symmetric/antisymmetric eigenstates | Packet oscillates between wells � quantum coherence; increase V0 to slow tunneling |
| 📊 Step Scatter | R+T=1; partial reflection even for E>V0 | See partial reflection from a step even when energy exceeds step height � purely quantum effect |
| 💥 WP Collision | interference fringes on overlap | Two packets meet; interference fringes appear at contact; fringes disappear after separation |
Curriculum Links
| Level | Topic | Covered |
|---|
| GCSE Physics | Wave-particle duality, photoelectric effect | Wave function concept, probability interpretation, quantisation |
| A-Level Physics | Quantum phenomena, de Broglie wavelength | ?=h/p; wave packets; qualitative tunneling; uncertainty principle |
| AP Physics C / Modern | Schr�dinger equation, harmonic oscillator | Gaussian eigenstates, energy quantisation, probability distributions |
| IB Physics HL | Wave mechanics, uncertainty principle | Heisenberg ?x?p=h/2; particle in a box; tunneling applications |
| University Physics / Maths | QM I & II: Schr�dinger equation, perturbation theory | Split-operator method, coherent states, WKB approximation, Dirac notation |