The Schrödinger equation
$i\hbar\frac{d}{dt}\left|\psi\left(t\right)\right\rangle =\hat{H}\left|\psi\left(t\right)\right\rangle$
Spin chain
Classical mechanics
$N$-dimensional phase space
Linear scaling with system size
Quantum mechanics
\( 2^N \)-dimensional Hilbert space
Exponential scaling with system size
Spin chain
$N=1$
Classical mechanics
Quantum mechanics
Spin chain
$N=2$
Classical mechanics
Quantum mechanics
Spin chain
$N=3$
Classical mechanics
Quantum mechanics
Spin chain
Classical mechanics
$N$-dimensional phase space
Linear scaling with system size
Quantum mechanics
\( 2^N \)-dimensional Hilbert space
Exponential scaling with system size
Spin chain
Classical mechanics
$N$-dimensional phase space
Linear scaling with system size
Quantum mechanics
\( 2^N \)-dimensional Hilbert space
Exponential scaling with system size
Big data - AI!
Neural networks can learn high dimensional space from samples
$256^{3}=16,777,216$ colors
$256^{2}=65,536$ pixels
$16,777,216^{65,536}=$ astronomical number
Neural networks can learn high dimensional space from samples
Works for wavefunctions too!
Ground state wavefunctions
At room temperature, most molecules are found in their ground state wavefunction, which has the lowest energy. This makes the ground state wavefunction a key focus in quantum chemistry.
$\frac{3}{2}k_{B}T\overset{\underbrace{T=298K}}{=}0.04\text{eV}\ll10.2\text{eV}$ $\left(\text{Excitation energy of H atom}\right)$
Finding the ground state wavefunction
Feynman's path integrals can be used to find the ground state wavefunction from any initial wavefunction via imaginary time evolution.
Finding the ground state wavefunction
Feynman's path integrals can be used to find the ground state wavefunction from any initial wavefunction via imaginary time evolution.
Finding the ground state wavefunction
Feynman's path integrals can be used to find the ground state wavefunction from any initial wavefunction via imaginary time evolution.
Finding the ground state wavefunction
Feynman's path integrals can be used to find the ground state wavefunction from any initial wavefunction via imaginary time evolution.
Finding the ground state wavefunction
Feynman's path integrals can be used to find the ground state wavefunction from any initial wavefunction via imaginary time evolution.
Finding the ground state wavefunction
Fermions
Fermions exhibit antisymmetry under exchange, meaning their wavefunction changes sign when swapped.
Store only the wavefunction where $x_{1}\leq x_{2}$
Complexity of $\mathscr{O}\left(N\log N\right)$
Fermions
Symmetrization by sorting!
Store only the wavefunction where $x_{1}\leq x_{2}$
Complexity of $\mathscr{O}\left(N\log N\right)$
Wigner crystallization
Thank You
- Bernheimer, L., Atanasova, H. & Cohen, G. Reports on Progress in Physics vol. 87 118001 (2024).
- Atanasova, H., Bernheimer, L. & Cohen, G. Nature Communications vol. 14 (2023).