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Quantum mechanics PHY3040
Time Evolution, two-level system
Dr Arkady Fedorov
PHYS3040 Semester 1, 2026. Workshop 3
1
Schrodinger Equation
• You saw the Schrodinger equation in PHYS2041
푑 휓 푡
푑푡
ꢀ
푖ℏ
= 퐻|휓 푡 〉
ꢀ
• 퐻 is the Hamiltonian operator: the generator of time evolution (or translation)
• Many textbooks treat this as a postulate
– It can be “derived” from Hamilton’s classical equations of motion, via the
Poisson bracket.
• It is not relativistically invariant. Generalise to:
– Klein-Gordon equation for bosons.
– Dirac equation for fermions. This predicts spin-½ intrinsic angular
momentum from purely mathematical considerations.
Hamiltonian
• We will be studying few-particle physics.
ꢀ
• Hilbert space (i.e. vector space with a scalar product) grows quickly with particle number, so 퐻 can be
complicated.
• But it has an energy eigenbasis
ꢀ
퐻 퐸, 푙, 푚, … = 퐸|퐸, 푙, 푚, … ⟩
with energy 퐸. There can be additional quantum labels 푙, 푚, etc
• You saw this for all the examples you solved last year.
−푅
• Recall the hydrogen atom (an electron orbiting a nucleus) had eigenstates |푛, 푙, 푚⟩, & 퐸ꢁꢂ =
1,2,3, … , ꢃ∞ = 13.6 푒푉, degeneracy 푙 = 0, … , 푛 − 1
∞ , 푛 =
2
ꢁ
– What do those letters stand for?
Schrodinger Equation preserves probability
ꢄ
ꢄ
2
• Evaluate
휓 푡
=
=
휓 푡 휓 푡
푑 휓 푡
푑푡
ꢄꢅ
ꢄꢅ
ꢀ
푖ℏ
= 퐻|휓 푡 〉
ꢄ
ꢄ
휓 푡
휓 푡 + 휓 푡
|휓 푡 ⟩
ꢄꢅ
ꢄꢅ
ꢀ
What condition on 퐻 guarantees conservation of probability?
Connection between SE and matrix mechanics
푑 휓 푡
푑푡
• Choose a basis 1 , 2 , 3 , …
ꢀ
푖ℏ
= 퐻|휓 푡 〉
σ
• Then 휓 푡 =
푐 푡 ꢆ for some coefficients 푐 푡
푗
푗
푗
• Evaluate Schrodinger Equation:
ꢇ
σ푗
• Finally get ODEs for the coefficients 푐푘 푡 = − ℏ
퐻푘푗푐푗 푡
Connection between SE and matrix mechanics
ꢇ
푑 휓 푡
푑푡
ꢀ
σ푗
• ODEs for the coefficients 푐푘 푡 = − ℏ
퐻푘푗푐푗 푡
푖ℏ
= 퐻|휓 푡 〉
• Write as a matrix
Solution of Schrodinger equation
• Simplest case: Schrodinger equation for diagonal Hamiltonian matrix
ꢀ
(i.e. ꢆ → 퐸 so that 퐻 퐸 = 퐸 퐸 , i.e. that ꢆ are energy eigenstates)
푗
푗
푗
푗
푖
풄 푡 = − 푯. 풄 푡
ℏ
퐸1
0 0
with 푯 =
0 퐸2
…
0
∶ …
• Solve this:
• Check normalisation
Solution of Schrodinger equation
푐 푡 = 푒−ꢇ ꢈ ꢅ/ℏ푐 0
ꢉ
• Solution is
푗
푗
퐸1
0 0
• In vector form:
푯 =
0 퐸2
…
0
∶ …
Solution of Schrodinger equation
푐 푡 = 푒−ꢇ ꢈ ꢅ/ℏ푐 0
ꢉ
• Solution is
푗
푗
퐸1
0 0
푯 =
0 퐸2
…
0
∶ …
푗 푒−ꢇ ꢈ ꢅ/ℏ푐 (0)| 퐸 ⟩
ꢉ
σ
• In Dirac notation:
휓 푡 =
푗
푗
Activity 1: Stationary states
푗 푒−ꢇ ꢈ ꢅ/ℏ푐 (0)| 퐸 ⟩
ꢉ
σ
In Dirac notation:
휓 푡 =
푗
푗
• What is evolution of state 휓 푡 if 휓 0 = 퐸 ?
푗
• What is the probability to find the system in the original state 휓 0 ?
PHYS3040 Semester 1, 2026. Workshop 3
Solution of Schrodinger equation
• General case
푖
흍 푡 = − 푯. 흍 푡
ℏ
ℎ11
ℎ21
ℎ31
ℎ12
ℎ22
∶
…
…
…
with 푯 =
−ꢇ푯ꢅ/ℏ
• How do we interpret 푒
?
Solution of Schrodinger equation
ꢀ
ꢀ
•
휓 푡 = 푈 푡 |휓 0 ⟩, where 푈 푡 is the Unitary evolution operator
ퟐ
ퟑ
−ꢇ푯 ꢅ/ℏ
−ꢇ푯 ꢅ/ℏ
3!
−ꢇ푯 ꢅ/ℏ
• Taylor series defines 푼 t = 푒
= 1 − 푖푯 푡/ℏ +
+
+ ⋯
2!
d
• Using the Taylor series, evaluate 푼 t . What differential equation does 푼 t satisfy?
dt
Solution of Schrodinger equation
ퟐ
ퟑ
−ꢇ푯 ꢅ/ℏ
−ꢇ푯 ꢅ/ℏ
−ꢇ푯 ꢅ/ℏ
• Taylor series defines 푼 t = 푒
= 1 − 푖푯 푡/ℏ +
+
+ ⋯
2!
3!
• Now the problem is to evaluate this. This is easy if H is already diagonal.
• If not, we should find the basis that diagonalises 푯 = 푷. 푫. 푷† where
– 푫 is the diagonal matrix of eigenvalues of 푯 and
– columns of 푷 are orthonormal eigenvectors of 푯, so 푷. 푷† = 푰
• Now evaluate Taylor series with 푯 = 푷. 푫. 푷†
Solution of Schrodinger equation
• 푯 = 푷. 푫. 푷† where
– 푫 is the diagonal matrix of eigenvalues of 푯 and
– columns of 푷 are orthonormal eigenvectors of 푯, so 푷. 푷† = 푰
−ꢇ푯 ꢅ/ℏ
• Then 푼 t = 푒
= 퐏. 푒−ꢇ푫 ꢅ/ℏ. 푷†
†
• Check 푼 t . 푼 t = 푰 (i.e. 푼 t is unitary):
Important results
• Schrodinger equation: 푖ℏ ꢄ ꢊ ꢅ
= 퐻 휓 푡 with 퐻= 퐻
ꢀ
ꢀ ꢀ†
ꢄꢅ
• In matrix/vector form: 흍 푡 = − ℏꢇ 푯. 흍 푡
ꢀ
• Solution to Schrodinger equation is 휓 푡 = 푈 푡 |휓 0 ⟩
ꢀ
−ꢇꢋꢅ/ℏ
−ꢇ푫 ꢅ/ℏ
. 푷† with 푈 = 푈
ꢀ
ꢀ−1
ꢀ†
• 푈 푡 = 푒
= 퐏. 푒
• “Diagonalising the Hamiltonian” is the main goal of theoretical quantum mechanics.
• You can get a Nobel prize for doing this (e.g. BCS theory, Kondo, Fractional Quantum Hall Effect,…)
Two-level Systems
• “2-level atom”, basis {|1⟩ , |2⟩}
• Some systems are exactly two-state systems
– E.g. an electron spin in a magnetic field;{ |푠푧 = −1/2⟩ , |푠푧 = +1/2⟩ }
• Others are approximations,
– e.g. Nitrogen position in an ammonia molecule
– e.g. Electron position in an ethylene molecule
• In both of these, there is a large spectrum of allowed states, but the important physics/chemistry depends
on the lowest energy states
PHYS3040 Semester 1, 2026. Workshop 3
16
Two-level atom: double well prototype
푥
• 푉 푥ො = 푉 푥ො + 푉푅(푥ො), where 푥ො is now an abstract coordinate
퐿
2
ො
푝
ꢀ
ꢎ
• 퐻 =
+ 푉(푥ො)
2ꢍ
• You solved the finite square well in PHYS2041, so these e/states
classically
form a basis.
• Physics “should” depend mostly on the lowest e/state for each
well: a localised state near each potential minimum
• We take the two states ꢏ and |ꢃ⟩ as a truncated basis
• Then
1
0
0
1
basis={ ꢏ =
,
ꢃ =
}
퐸퐿 + 퐸푅
퐸퐿 − 퐸푅
퐸퐿
Δ
ꢀ
퐻 =
=
핀 +
휎푧 + Δ 휎ꢐ
0.0
- 0.2
- 0.4
- 0.6
- 0.8
- 1.0
0.0
- 0.2
- 0.4
- 0.6
- 0.8
- 1.0
Δ 퐸푅
2
2
• This is the same mathematical form as for a spin ½
- 4
- 2
0
2
4
- 4
- 2
0
2
4
x
x
PHYS3040 Semester 1, 2026. Workshop 3
17
Eigenstates and energy gap of a double well potential
푗 푒−ꢇ ꢈ ꢅ/ℏ푐 (0)| 퐸 ⟩. So we need the eigenvalues 퐸 and eigenvectors
ꢉ
ꢀ
σ
• Recall 휓 푡 = 푈 푡 |휓 0 ⟩ or 휓 푡 =
푗
푗
푗
|퐸 〉.
푗
퐸퐿
Δ
ꢀ
• Find eigenvalues of 퐻 =
by solving characteristic equation det 퐴 − 휆 퐼 = 0.
Δ 퐸푅
• Check your solution if Δ = 0. Does it make sense?
• Check your solution if 퐸퐿 = 퐸푅 = 퐸0. Does it make sense? What symmetry does this case correspond to?
• What is the gap (difference between ground and excited state)?
PHYS3040 Semester 1, 2026. Workshop 3
18
Avoided crossing
2
2
ꢈ +ꢈ ±
ꢈ −ꢈ
ꢑ ꢒ
+4Δ
ꢑ
ꢒ
• Eigenvalues are
. Set 퐸푅 = 0
2
6
4
2
“Avoided crossing”
“Landau-Zener crossing”
2 Δ
0
- 2
- 4
- 6
- 6
- 4
- 2
0
2
4
6
EL /Δ
PHYS3040 Semester 1, 2026. Workshop 3
19
Activity 1: Symmetric well. Eigenstates
0 Δ
ꢀ
Consider special case where 퐸퐿 = 퐸푅 = 0: 퐻 =
Δ 0
1
1
1. Check that ± =
are e/vecs and find corresponding eigenvalues
2
±1
1
0
0
2. Write down ꢏ =
and ꢃ =
as a linear combination of ± .
1
PHYS3040 Semester 1, 2026. Workshop 3
20
Activity 2:Symmetric well. Evolution
푗 푒−ꢇ ꢈ ꢅ/ℏ푐 (0)| 퐸 ⟩(*)
ꢉ
σ
• Recall 휓 푡 =
푗
푗
• Suppose initial state is 휓(0) = ꢏ . What are 퐸 and 푐 (0)?
푗
푗
• Compute 휓 푡 using (*). Eliminate 퐸 by expressing them in terms of |ꢏ〉 and |ꢃ〉. Compute probability
푗
to find system in ꢏ .
PHYS3040 Semester 1, 2026. Workshop 3
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Probability to find the state in one of the wells
•
휓 푡 = cos(Δ푡) ꢏ − 푖 sin(Δ푡) ꢃ
• Probability to be in the initial state
2
|
| 휓 0 휓 푡
푡
• “Coherent oscillations”
• Later we will discuss “decoherence”: oscillations decay over time.
• We have now “solved” the evolution for a time-independent Hamiltonian for a 2-level system
PHYS3040 Semester 1, 2026. Workshop 3
22
Consider trajectory of a state on a Bloch sphere
Particle tunnels through the barrier
ꢇꢓ
휓 = cos 휃 ꢏ + 푒 sin 휃 |ꢃ〉
z
|ꢏ⟩
휓
휃
휙
y
x
|ꢃ⟩
time
PHYS3040 Semester 1, 2026. Workshop 3
23
Spin precesses around the constant field
Particle tunnels through the barrier
휓 푡 = cos Δ푡 ↑푧 − 푖 sin Δ푡 ↓푧
z
| ↑푧⟩
휃
휙
y
x
| ↓푧⟩
time
PHYS3040 Semester 1, 2026. Workshop 3
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Some history
• Quantum oscillation is direct evidence of QM
• Would be not so surprising for electrons but harder to
observe
• Some large systems can theoretically exhibit quantum
tunnelling (superconducting SQUID loops)
• Observation of these oscillations would be a direct evidence
of QM at the macroscopic level
PHYS3040 Semester 1, 2026. Workshop 3
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Some history
• Caldeira and Leggett studied this question back in
1983
• They formalized description of dissipation
(decoherence) as a cause of not being able to
observe these oscillations
>4000 citations
PHYS3040 Semester 1, 2026. Workshop 3
26
Some history
• Superconducting ring has a double well potential in the
phase space
• Superconducting ring has two quantum states with
macroscopic persistent currents (clock and counter clock
directions)
• Can be controlled externally and detected with a
magnetometer
• Fabricated in the cleanroom
A. Fedorov, P. Macha, A.K. Feofanov, C. Harmans, J.E. Mooij, Physical Review Letters 106 (17), 170404
PHYS3040 Semester 1, 2026. Workshop 3
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Some history
• Experimental protocol:
– Bias wells to prepare the system in the left well
– Making well symmetric and make the system to
evolve
– Bias wells again to measure if the system is still in
the left well
• Tunnelling rate can be also controlled
• Oscillations are decaying. Why?
A. Fedorov, P. Macha, A.K. Feofanov, C. Harmans, J.E. Mooij, Physical review letters 106 (17), 170404
PHYS3040 Semester 1, 2026. Workshop 3
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Summary
• 2-level atoms are simple, but capture important physics
– Avoided level crossing is signature of tunneling
– Coherent oscillation between eigenstates (can be observed in a variety of systems including some
close to macroscopic ones)
– Both are signs of the same physical phenomenon: coupling of two-states
Next : driven two-level system and Rabi oscillations.
PHYS3040 Semester 1, 2026. Workshop 3
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