Introduction to computational quantum mechanics
Lecture 1: From classical to quantum mechanics
Simen Kvaal
simen.kvaal@cma.uio.no
Centre of Mathematics for Applications
University of Oslo
Seminar series in quantum mechanics at CMA
Fall 2009
Outline
A recap of classical Hamiltonian dynamics
From classical mechanics to quantum mechanics (QM)
Wave equation nature of Schrödinger equation
A simple example: particle in infinite grave
Conclusion
Outline
A recap of classical Hamiltonian dynamics
From classical mechanics to quantum mechanics (QM)
Wave equation nature of Schrödinger equation
A simple example: particle in infinite grave
Conclusion
Material systems
Rd
I
I
I
complete state
We consider a system of n particles of mass m, living in Rd
Each particle has coordinate ~xj ∈ Rd , j = 1, · · · , n and momentum
~pj ∈ Rd .
At time t ∈ R, the state of the material system is entirely given by
(x, p) = (~x1 , · · · ,~xn ,~p1 , · · · ,~pn ) ∈ R2nd
I
R2nd
(phase space).
Measurable quantities are functions ω(x, p) of the coordinates and
momenta.
Hamilton’s equation(s) of motion
I
The most important observable (besides ~xj and ~pj ) is the Hamiltonian:
n
1
k~pj k2 + V(~x1 , · · · ,~xn )
2m
j=1
H(x, p) = ∑
I
I
(This is just a very typical example)
The evolution of the system is given by Hamilton’s equation(s) of
motion:
~x˙ j
= ∇~pj H(x, p) (= ~pj /m in ex.)
~p˙ j
= −∇~qj H(x, p)
In the example, we obtain:
m~x¨ j = −∇~xj V(~x1 , · · · ,~xn ),
i.e., Newton’s second law
The mechanistic world view and “Laplace’ demon”
We may regard the present state of the universe as the effect of its
past and the cause of its future. An intellect which at a certain
moment would know all forces that set nature in motion, and all
positions of all items of which nature is composed, if this intellect
were also vast enough to submit these data to analysis, it would
embrace in a single formula the movements of the greatest bodies
of the universe and those of the tiniest atom; for such an intellect
nothing would be uncertain and the future just like the past would
be present before its eyes.
— Pierre-Simon Laplace
Simple example
We specialize to a single particle (n = 1) in one space dimension:
I Hamiltonian:
1 2
H(x, p) =
p + V(x)
2m
I Hamilton’s equation of motion:
ẋ = p/m,
I
ṗ = −V 0 (x).
We define E(t) = H(x(t), p(t)), and easily calculate:
Ė ≡
∂H
= 0,
∂t
i.e., conservation of energy. This holds in general whenever ∂H/∂t = 0.
Visualization
We may visualize the motion as “roller coaster motion” where V(x) is the
elevation of the track: In a vertical gravity field, a particle sliding on the
curve {(x, V(x))} experiences a net acceleration proportional to −V 0 (x), as
in our example.
Outline
A recap of classical Hamiltonian dynamics
From classical mechanics to quantum mechanics (QM)
Wave equation nature of Schrödinger equation
A simple example: particle in infinite grave
Conclusion
Before we begin . . .
I
I
I
We will now outline a somewhat simplified version of non-relativistic
QM for n particles.
The “correct” version includes topics we will return to in due time.
(The pauli principle for fermions, spin, usw.)
Some (physical) topics are omitted: Theory of measurements, collapse
of the wave function, and so on. We will probably fill in these details in
a later lecture, as they are not essential to us at the moment.
States in quantum mechanics
I
I
At time t, the phase space point (x, p) is replaced by a wavefunction
ψ(~x1 , · · · ,~xn ), in a (complex) Hilbert space H , with unit norm kψk = 1.
We will return to H in the next lecture, but in our simplified QM,
H = L2 (Rnd )
I
Furthermore,
|ψ(~r1 , · · · ,~rn , t)|2 = probability density of locating particles
I
Any ψ ∈ H is a legal quantum state; “superposition principle”.




ψleft = 

ψright = 

States in quantum mechanics
I
I
At time t, the phase space point (x, p) is replaced by a wavefunction
ψ(~x1 , · · · ,~xn ), in a (complex) Hilbert space H , with unit norm kψk = 1.
We will return to H in the next lecture, but in our simplified QM,
H = L2 (Rnd )
I
Furthermore,
|ψ(~r1 , · · · ,~rn , t)|2 = probability density of locating particles
I
Any ψ ∈ H is a legal quantum state; “superposition principle”.

1
√ ψleft + ψright = 
2


Observables
I
I
Observables ω are replaced by self-adjoint linear operators Ω on H
Important examples:
I
Position operator / multiplication with (~xj )k = xjk :
(x̂jk ψ)(x) = xjk ψ(x),
I
Momentum operator / differentiation:
(p̂jk ψ)(x) = −ih̄
I
I
x := (~x1 , · · · ,~xn )
∂
ψ(x),
∂xjk
h̄ = fundamental physical const.
Observables usually have no definite value! In this sense, QM is no
longer deterministic.
One speaks instead of, e.g., the expectation value, variance etc:
hΩi = hψ, Ω, ψi ,
σ2Ω = hΩ2 i − hΩi2 ,
...
Example
I
I
Consider n = d = 1, i.e., H = L2 (R).
Observable: Ω = x̂:
hx̂i = hψ, x̂ψi =
I
Z
ψ(x)xψ(x) dx =
Z
xP(x) dx.
Observable: Ω = T = p̂2 /2m (kinetic energy):
hTi = hψ, Tψi = −
h̄2
2m
Z
ψ(x)ψ00 (x) dx =
h̄2
2m
Z
|ψ0 (x)| dx.
The quantum Hamiltonian
I
In general, observables are obtained by substituting x and p by their
operators x̂ and p̂:
Ω = ω(x̂, p̂).
I
In particular, the Hamiltonian becomes:
n
h̄2 2
1
k~pj k2 +V(~x1 , · · · ,~xn ) −→ Ĥ = − ∑
∇j +V(~x1 , · · · ,~xn )
j=1 2m
j=1 2m
n
H(x, p) = ∑
Here,
∇2j =
d
k=1
I
∂2
∑ ∂x2
= Laplacian with respect to ~xj .
jk
Of course, the self-adjointedness of Ĥ depends on V.
The Schrödinger equation
I
The time evolution of the quantum state ψ(t) is governed by the
Schrödinger equation:
−ih̄ψ̇(t) = Ĥψ(t)
I
Given Ĥ ∗ = Ĥ, this equation always has a solution:
ψ(t) = U(t)ψ(0) = e−itĤ/h̄ ψ(0),
I
The propagator U(t) is a one-parameter unitary group:
U(t + s) = U(t)U(s),
I
∀ψ(0) ∈ H
U(t)∗ = U(t)−1 = U(−t),
U(0) = 1.
If Ĥ depends explicitly on t, the existence of dynamics is more subtle.
Postulate 1
Postulate 1: States and observables
The state of a quantum system at a given time t ∈ R is uniquely determined
(up to normalization) by an element ψ(t) in a (complex) Hilbert space H .
Observables – quantities we may measure in experiments – are represented
by self-adjoint operators Ω.
Postulate 2
Postulate 2: Time evolution
The time-evolution of the state ψ(t) is governed by the Schrödinger
equation, viz,
−ih̄ψ̇(t) = Ĥψ(t)
This is a wave equation, describing “matter waves”.
Postulate 3
Postulate 3: Measurements
The only possible outcomes of a measurement of an observable Ω is
elements ω ∈ σ(Ω) = σp ∪ σc , i.e., the spectrum of Ω.
For a (simple) eigenvalue ω ∈ σp (Ω) the probability of the outcome ω is
P(ω) = hφω , ψi ,
where φω is the eigenvector.
For an approximate eigenvalue ω ∈ σc (Ω),
P(ω)dω = lim h(φω )n , ψi dω
n→∞
is the probability density, where {(φω )n } is a Weyl sequence for the
approximate eigenvalue.
Outline
A recap of classical Hamiltonian dynamics
From classical mechanics to quantum mechanics (QM)
Wave equation nature of Schrödinger equation
A simple example: particle in infinite grave
Conclusion
Free particle on the real line
I
I
We consider n = d = 1 ⇒ H = L2 (R)
Free particle, i.e., it travels without influence:
H=
I
p2
h̄2 ∂2
=−
2m
2m ∂x2
p
t,
m
p ≡ p(0) = const,
E≡
(Approximate) eigenfunction of H:
φp (x) = eipx/h̄ ,
I
kinetic energy
Classical solution is linear motion, naturally:
x(t) = x(0) +
I
≡
Ep =
p2
2m
Time evolution:
p
p2
e−itH/h̄ φp (x) = ei( h̄ x− 2mh̄ t) = ei(kx−ωt)
where k = p/h̄. Plane wave!
p2
2m
Free particle on the real line 2
I
Let’s try a nontrivial initial condition, a wave packet:
p (x − x0 )2
0
exp
i x .
ψ(x) = N exp −
4σ2
h̄
I
Some statistics:
hx̂i = x0 ,
I
I
σx = σ (!),
hp̂i = p0 .
Localized quantum particle centered at x0 with “width” σ travelling
with momentum p0 .
One can show that U(t)ψ(x) has the same functional form, but with:
x0 (t) = x0 + pt/m,
σ0 (t) = getting larger
Outline
A recap of classical Hamiltonian dynamics
From classical mechanics to quantum mechanics (QM)
Wave equation nature of Schrödinger equation
A simple example: particle in infinite grave
Conclusion
Formulation of problem
I
We consider a single particle in d = 1 spatial dimensions, restricted to
exist on the interval 0 < x < 1. Thus,
H = L2 (0, 1)
I
For simplicity, we consider the Hamiltonian
H=
I
1 2
h̄2 ∂2
p =−
= kinetic energy only.
2m
2m ∂x2
The Schrödinger equation reads (h̄ = m = 1):
−i
∂
1 ∂2
ψ(x, t) = −
ψ(x, t)
∂t
2 ∂x2
with boundary conditions ψ(0, t) = ψ(1, t) = 0.
Solution
I
I
I
I
Fact: H is self-adjoint on Sobolev space H 2 (0, 1) with compact inverse.
A sequence of real eigenvalues E1 < E1 < · · · % ∞.
Eigenvalue problem:
1
− φ00n (x) = Eφn (x).
2
Solution:
√
n2 π2
φn (x) = 2 sin(nπx), En =
2
Decomposition of ψ(x) in eigenfunctions ⇔ Fourier sine series:
ψ(x) =
∞
∑ hφn , ψi φn (x).
n=1
I
Solution of Schrödinger equation:
ψ(x, t) =
∞
∑ hφn , ψi e−itEn φn (x)
n=1
Animated particle in grave
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I
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Ititial condition 1: Eigenfunction of E1 ; “ground state”, then with a
mixture.
Initial condition 2: Wave packet.
Initial condition 3: Constant function.
Outline
A recap of classical Hamiltonian dynamics
From classical mechanics to quantum mechanics (QM)
Wave equation nature of Schrödinger equation
A simple example: particle in infinite grave
Conclusion
Next time . . .
I
I
I
I
The time-independent Schrödinger equation: the eigenvalue problem
for H
A bestiary of Hamiltonians: from chemistry to nuclear physics
Elements of spectral theory
Proof of the existence of dynamics