Quantum Physics Tutorial 1 focuses on fundamental problems in quantum mechanics, including kets, bras, scalar products, and the properties of Hermitian operators. It provides detailed exercises for students studying quantum physics concepts, such as orthonormal states and projection operators. This tutorial is ideal for undergraduate students in physics or anyone preparing for exams in quantum mechanics. The document includes various problems and solutions that enhance understanding of quantum formalism and mathematical techniques.

Key Points

  • Explores the properties of kets and bras in quantum mechanics.
  • Includes exercises on scalar products and orthonormal states.
  • Discusses the hermiticity of operators in quantum physics.
  • Provides problems related to projection operators and their normalization.
computer science
1 page
Language:English
Type:Tutorial
Biology
computer science
1 page
Language:English
Type:Tutorial
Biology
272
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Quantum Physics tutorial 1
January 2026
Problems
1. (Note: We will see later in the lectures that kets are represented by column matrices and
bras by row matrices; this example is offered earlier than it should because we need to show
some concrete illustrations of the formalism.) Consider the following two kets:
|ψ⟩ =
−3i
2 + i
4
|ϕ⟩ =
2
−i
2 − 3i
(a) Find the bra ⟨ϕ|.
(b) Evaluate the scalar product ⟨ϕ | ψ⟩
(c) Examine why the products |ψ⟩ |ϕ⟩ and ⟨ϕ| ⟨ψ| do not make sense.
2. Consider the states |ψ⟩ = 3i |ϕ
1
⟩ − 7i |ϕ
2
⟩ and |χ⟩ = − |ϕ
1
⟩ + 2i |ϕ
2
⟩, where |ϕ
1
⟩ and |ϕ
2
⟩
are orthonormal.
(a) Calculate |ψ + χ⟩ and ⟨ψ + χ|.
(b) Calculate the scalar products ⟨ψ | χ⟩ and ⟨χ | ψ⟩. Are they equal?
(c) Show that the states |ψ⟩ and |χ⟩ satisfy the Schwarz inequality.
(d) Show that the states |ψ⟩ and |χ⟩ satisfy the triangle inequality.
3. Consider the two states |ψ
1
⟩ = 2i|ϕ
1
⟩ + |ϕ
2
⟩ − a|ϕ
3
⟩ + 4|ϕ
4
⟩ and |ψ
2
⟩ = 3|ϕ
1
⟩ − i|ϕ
2
⟩ + 5|ϕ
3
⟩ −
|ϕ
4
⟩, where |ϕ
1
⟩, |ϕ
2
⟩, |ϕ
3
⟩ and |ϕ
4
⟩ are orthonormal kets, and where a is a constant. Find
the value of a so that |ψ
1
⟩ and |ψ
2
⟩ are orthogonal.
4. (a) Discuss the hermiticity of the operators
ˆ
A +
ˆ
A
†
, i
ˆ
A +
ˆ
A
†
, and i
ˆ
A −
ˆ
A
†
.
(b) Find the Hermitian adjoint of f(
ˆ
A) =
(
1+i
ˆ
A+3
ˆ
A
2
)(
1−2i
ˆ
A−9
ˆ
A
2
)
5+7
ˆ
A
.
(c) Show that the expectation value of a Hermitian operator is real and that of an anti-
Hermitian operator is imaginary.
5. Show that the operator |ψ⟩⟨ψ| is a projection operator only when |ψ⟩ is normalized.
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FAQs

What are the two kets defined in the Quantum Physics tutorial?
The two kets defined in the tutorial are |ψ⟩ and |ϕ⟩. Specifically, |ψ⟩ is represented as a column matrix with elements (-3i, 2 + i, 4), while |ϕ⟩ is represented as (2, -i, 2 - 3i). These kets serve as foundational examples for exploring quantum states and their properties.
How do you evaluate the scalar product ⟨ϕ | ψ⟩?
To evaluate the scalar product ⟨ϕ | ψ⟩, you take the bra ⟨ϕ|, which is the conjugate transpose of the ket |ϕ⟩, and multiply it by the ket |ψ⟩. This involves performing matrix multiplication on the respective components of the kets, resulting in a complex number that represents the inner product of the two states.
What condition must be met for the operator |ψ⟩⟨ψ| to be a projection operator?
The operator |ψ⟩⟨ψ| is a projection operator only when the ket |ψ⟩ is normalized. This means that the inner product ⟨ψ | ψ⟩ must equal 1. If |ψ⟩ is not normalized, the operator will not satisfy the properties required for it to act as a projection operator in quantum mechanics.
What is the Schwarz inequality in the context of the states |ψ⟩ and |χ⟩?
The Schwarz inequality states that for any two quantum states |ψ⟩ and |χ⟩, the absolute value of their inner product is less than or equal to the product of their norms. In the tutorial, it is shown that the states |ψ⟩ = 3i|ϕ1⟩ - 7i|ϕ2⟩ and |χ⟩ = -|ϕ1⟩ + 2i|ϕ2⟩ satisfy this inequality, confirming the relationship between the states in terms of their magnitudes and inner products.
What is the significance of Hermitian operators in quantum mechanics?
Hermitian operators are significant in quantum mechanics because they have real eigenvalues, which correspond to measurable quantities. The tutorial discusses the hermiticity of operators such as Aˆ + Aˆ† and i(Aˆ + Aˆ†), highlighting their properties and implications in physical measurements. Understanding these operators is crucial for analyzing quantum systems and ensuring that the results of measurements are physically meaningful.
How can you find the value of a for orthogonality of |ψ1⟩ and |ψ2⟩?
To find the value of a that ensures the orthogonality of the states |ψ1⟩ and |ψ2⟩, you need to compute the inner product ⟨ψ1 | ψ2⟩ and set it equal to zero. The states are defined as |ψ1⟩ = 2i|ϕ1⟩ + |ϕ2⟩ - a|ϕ3⟩ + 4|ϕ4⟩ and |ψ2⟩ = 3|ϕ1⟩ - i|ϕ2⟩ + 5|ϕ3⟩ - |ϕ4⟩. By expanding this inner product and solving for a, you can determine the specific value that makes the states orthogonal.

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