Linear vector space; Quiz I (Hidar 27/2017)

Which of the following is NOT a property of a vector space?

• Associativity of vector addition

• Commutativity of scalar multiplication

• Distributivity of scalar multiplication over vector addition

• Existence of a zero vector

Answer explanation

Commutativity of scalar multiplication is not a property of vector spaces. In vector spaces, scalar multiplication is associative and distributive, but it does not require commutativity.

What is the dimension of the vector space spanned by the vectors (1, 0, 0), (0, 1, 0), and (1, 1, 1)?

• 1

• 2

• 3

• 4

Answer explanation

The vectors (1, 0, 0) and (0, 1, 0) are linearly independent and span a plane. The vector (1, 1, 1) is not in that plane, adding a third dimension. Thus, the dimension of the vector space spanned by these vectors is 3.

Two vectors are linearly independent if

• • One is a scalar multiple of the other

• • Their dot product is zero

• • They cannot be expressed as a linear combination of each other

• • Their cross product is zero

Answer explanation

Two vectors are linearly independent if they cannot be expressed as a linear combination of each other. This means neither vector can be written as a scalar multiple of the other, which is the correct choice.

The state of a quantum system is represented by:

• A real number

• A complex number

• A vector in a complex Hilbert space

• A matrix

Answer explanation

The state of a quantum system is represented by a vector in a complex Hilbert space, which encapsulates all possible states and their probabilities. This is fundamental to quantum mechanics, distinguishing it from classical systems.

The time evolution of a quantum system is governed by:

• Newton's laws of motion

• The Schrödinger equation

• The Heisenberg uncertainty principle

• The Born rule

Answer explanation

The time evolution of a quantum system is described by the Schrödinger equation, which provides a mathematical framework for predicting how quantum states change over time, unlike the other options which pertain to different aspects of physics.

Two operators, A and B, commute if:

• AB = BA

• AB + BA = 0

• AB - BA = 1

• AB = 0

Answer explanation

Two operators A and B are said to commute if their product is independent of the order in which they are applied, which is mathematically expressed as AB = BA. This means the correct choice is AB = BA.

A Hermitian operator is:

• Equal to its complex conjugate

• Equal to its transpose

• Equal to its adjoint

• None of the above

Answer explanation

A Hermitian operator is defined as one that is equal to its adjoint (or conjugate transpose). This means it satisfies the condition A = A†, making 'Equal to its adjoint' the correct choice.

Eigenvalues of a Hermitian operator are always:

• Real

• Imaginary

• Complex

• Undetermined

Answer explanation

Eigenvalues of a Hermitian operator are always real due to the properties of Hermitian operators in quantum mechanics, which ensure that their eigenvalues correspond to observable quantities.

The eigenvalue equation for an operator A is:

• Aψ = λψ

• Aψ = 0

• ψA = λψ

• ψA = 0

Answer explanation

The eigenvalue equation for an operator A is correctly represented as Aψ = λψ, where A acts on the eigenfunction ψ to yield a scalar multiple (λ) of that function. This defines the relationship between the operator and its eigenvalues.

The commutator of the position and momentum operators is:

• -

• iħ

• -iħ

• ħ²

Answer explanation

The commutator of the position operator \(\hat{x}\) and momentum operator \(\hat{p}\) is given by \([\hat{x}, \hat{p}] = \hat{x}\hat{p} - \hat{p}\hat{x} = i\hbar\). Thus, the correct answer is \(i\hbar\).