what makes quantum physics different from earlier …quantum chemistry group what makes quantum...
TRANSCRIPT
Quantum Chemistry Group
What makes Quantum Physics different from earlier Physics?
23 January 2019
Seminar at IMAC, UJI
Juan I. Climente
Quantum physics vs classical physics
1. The experiments
2. The theory
3. Bonus track
Quantum physics vs classical physics
1. The experiments
2. The theory
3. Bonus track
4
An experiment with bullets
Machinegun
1
2
detector
x
P
P12
P1
P2
a) Detection in lumps (discrete clicks, identical loudness)
b) P12
= P1 + P
2
Particle signatures
5
An experiment with waves
Wavesource
1
2
detector
x
I
I1
a) Detection continuous (variable loudness)
b) I12
=|H1 +H
2|2 ≠ I
1+I
2 Wave signatures
I2
I12
6
An experiment with electrons
1
2
detector
x
P
P1
a) Detection in lumps (discrete clicks, identical loudness)
b) I12
=|ψ1 +ψ
2|2 ≠ P
1+P
2
Dual nature:particle-wave
P2
P12
Electron gun
7
Let's watch the electrons
1
2
detector
x
PP
1
'
P2
' '
Electron gun
P12
'
«It is impossible to design an apparatus to determine which of two alternatives is taken without destroying the interference pattern»
Heisenberg's Uncertainty Principle
8
Quantum physics vs classical physics
Summary of first principles:
(1) The probability of an event is P=|ψ|2, where ψ is a complex number we call amplitude.
(2) If an event can take place in different ways, then P=|∑
i ψ
i|2.
(3) If an experiment is able of determining which alternative is taken, then P=∑
i P
i .
Why?
We only talk about predicting odds
Seems to explain everything on atomic scale
9
Quantum physics vs classical physics
1. The experiments
2. The theory
3. Bonus track
10
Schrödinger’s Equation
1900-1926 – The years of confusion
1927 – Postulate of the differential equation
Newton’s 2nd law- particles -
Wave equation- waves -
Planck’s equation
De Broglie’s hypothesisHeisenberg’s uncertainty
let’s make itdual
11
Schrödinger’s Equation
Newton’s 2nd law- particles -
Wave equation- waves -
Relativistic particles:
Slow particles:
Guess:
12
Schrödinger’s Equation
Physical-mathematical considerations:
1) ψ(r,t) is the (complex) amplitude associated with the probability of finding a particle at (r,t). Call it wave function:
2) Total probability of finding the particle confined in a volume must be unity.
normalization condition
Then, ψ(r,t) must be square integrable. For bound particles, if r→∞ then ψ(r,t)→0.
3) ψ(r,t) must be well behaved.
3.1) Finite. 3.2) Single-valued. 3.3) Continuous. 1st derivatives too.
4) Trivial solution, meaningless.
13
Let’s play!
Particle in a 1D box
xTime-independent
Schrödinger Equation - stationary states -
L
Boundary conditions
14
It’s a quantum world
Energy is quantised
Quantised energies explain discrete atomic spectra:
H Balmer series, 1885
15
Operators
Classical physics:
Quantum physics:
Schrödinger Equation
Remember: We only talk about predicting odds
Mean value of a physical magnitude:
All operators are Hermitian
Rule to movefrom classical to quantum
16
Variational principle
Let ψ be the eigenfunction of the Hamiltonian, with eigenvalue Eexact
.
For any function Φ with the same boundary conditions, and well-behaved, it holds:
, define an
and obtain the coefficients by
It is customary to build a basis set formed by linearly independent functions:
whose scalar product is given by
minimizing
Optimizing the basis set to reduce the required Hilbert space dimension has beenthe front line of Quantum Chemistry research for over 40 years now.
aproximate solution as
17
Quantum Physics and Mathematics:
( * ) Algebra of creation/annihilation operators
( * ) Linear algebra
( * ) Group theory (symmetry and permutation groups)
( * ) Numerical methods
18
What makes Quantum Physics different from earlier Physics?
Seminar at IMAC, UJI
(2) No longer «deterministic». We calculate probable values
(1) Dual character particle-wave
(4) Schrödinger equation replaces Newton 2nd law & wave eq.
(5) Physical magnitudes have quantised values
(3) Simultaneous knowledge of some magnitudes impossible
19
Quantum physics vs classical physics
1. The experiments
2. The theory
3. Bonus track
20
Let ψ be the eigenfunction of the Hamiltonian, with eigenvalue Eexact
.
For any function Φ with the same boundary conditions, and well-behaved, it holds:
Variational Quantum Montecarlo
We choose a trial function Φα with a set of adjustable parameters α={α
1,α
2,α
3...},
and look for those which minimize the energy.
An efficient way must be found to calculate the multi-dimensional integral above.
Stochastic methods: draw random values within V,average integrand, minimize energy or variance
21
Variational Quantum Montecarlo
Smart sampling critical for efficient calculation
Random sampling (between -1 and 1) Metropolis algorithm: priority to points with largest |Φ|2 values. Good results with much fewer points.
Standard technique for many-electron systems, but fails for charges of opposite sign.
e- h+