QM II Tutorial 1
PHYS3040 Quantum Physics
Tutotial 1 Week 2 2026
Problem 1
What is the definition of the orthonormaland complete basis? Is this basis unique? Explain the
similarities and differences between a complete orthonormal basis in Euclidean and Hilbert
spaces.
Problem 2
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Consider the state 휓 = (3|0⟩ + 푖|1⟩)/√10 where {|0⟩, |1⟩} is an
orthonormal complete
basis.| ⟩
1. Show that
orthonormality of the basis.
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휓
is normalised using properties of the inner product and the
2. Write down the state 휓 in a vector form.
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3. Show that 휓 is normalised using matrix multiplication and its vector form.
Problem 3
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| ⟩
Consider states ± = ( 0 ± |1⟩)/√2 .
1. Show that they are orthogonal and normalised.
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2. Write down state 휓 = (|0⟩ + 3푖|1⟩)/√10 as a liner superposition of ± .
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3. Using ± as a new basis. Write down the vector form of 휓 in the new basis and show
that it is normalised using matrix mechanics in this basis.
Problem 4
In Workshop 2 you will learn that any physical state can be represented as vector on a Block
ꢀ
ꢀ
ꢁ휙
| ⟩
휓 = cos (2) |0⟩ + sin (2) 푒
sphere using two real parameters {휑, 휃} as
|1⟩.
PHYS3040 Quantum Physics
Tutotial 1 Week 2 2026
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| ⟩
1. Show the position of a vector on the Bloch sphere for states: ± = ( 0 ± |1⟩)/√2 and
| ⟩
휓 = (|0⟩ + 2푖|1⟩)/√5.
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