SME
Quantum Mechanics · Problem type
Measuring the energy of a superposition
Use this for “A particle in a box is in the state 3ψ₁ + 4ψ₂, with E₁ = 0.2 eV. What energies can a measurement give, with what chances, and what is ⟨E⟩?”
Chart
The bars add to 1.
E(X) = 0.376 × 0.5 + 1.504 × 0.5
E(X) = 0.94
This is an example. Type your own values to start a new problem.
ψ
=
c₁
ψ₁
+
c₂
ψ₂
⟨E⟩
=
P₁
×
E₁
+
P₂
×
E₂
=
⟨E⟩ (eV)
Formulas
P₁
=
c₁
2
c₁
2
+
c₂
2
0.5
=
1
2
1
2
+
1
2
P₂
=
c₂
2
c₁
2
+
c₂
2
0.5
=
1
2
1
2
+
1
2
P₁
+
P₂
=
1
0.5 + 0.5 = 1
⟨E⟩
=
(1
+
3
×
P₂
)
×
E₁
0.94 = (1 + 3 × 0.5) × 0.376
E₂
=
4
×
E₁
1.504 = 4 × 0.376
⟨E⟩
=
P₁
×
E₁
+
P₂
×
E₂
0.94 = 0.5 × 0.376 + 0.5 × 1.504
Assumptions
•
The state is c₁ψ₁ + c₂ψ₂, the box’s two lowest states, so E₂ = 2²E₁ = 4E₁; the amplitudes are real.
•
The amplitudes need not be normalized: the page divides by c₁² + c₂², so Pᵢ = cᵢ² ÷ (c₁² + c₂²).
•
One measurement gives E₁ or E₂, never ⟨E⟩; ⟨E⟩ is the average over many copies of the state.
Step-by-step
WE KNOW
c₁ = 1, c₂ = 1, E₁ = 0.376 eV
FIND
chance of E₁ (P₁), chance of E₂ (P₂), second energy (E₂), expected energy (⟨E⟩)
Step 1 · Find chance of E₁ (P₁)
Use P₁ = c₁² ÷ (c₁² + c₂²)
Born’s rule: the chance is the amplitude squared, over the sum of the squares so the chances add to 1.
P₁
=
1
2
1
2
+
1
2
P₁
=
1
2
1 + 1
P₁
=
1
2
2
P₁
=
1
2
P₁
=
0.5
Step 2 · Find chance of E₂ (P₂)
Use P₂ = c₂² ÷ (c₁² + c₂²)
The same for ψ₂: its amplitude squared over the sum of the squares.
P₂
=
1
2
1
2
+
1
2
P₂
=
1
2
1 + 1
P₂
=
1
2
2
P₂
=
1
2
P₂
=
0.5
Step 3 · Find second energy (E₂)
Use E₂ = 4 × E₁
Box
levels
go
as
n
2
:
the
second
is
2
2
=
4
times
the
first.
E₂
=
4
×
0.376
E₂
=
1.504
eV
Step 4 · Find expected energy (⟨E⟩)
Use ⟨E⟩ = P₁ × E₁ + P₂ × E₂
Weight each energy a measurement can give by its chance, then add.
⟨E⟩ = 0.5 × 0.376 + 0.5 × 1.504
⟨E⟩ = 0.188 + 0.752
⟨E⟩ = 0.94 eV
Check
Put the numbers back in. Both sides should match.
0.5
=
1
2
1
2
+
1
2
✓
0.5
=
1
2
1
2
+
1
2
✓
0.5 + 0.5 = 1 ✓
0.94 = (1 + 3 × 0.5) × 0.376 ✓
1.504 = 4 × 0.376 ✓
0.94 = 0.5 × 0.376 + 0.5 × 1.504 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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