4_vector_spaces_QM.md

title: Vector spaces in quantum mechanics

4. Vector spaces in quantum mechanics

The lecture on vector spaces in quantum mechanics consists of the following parts:

• 4.1. Dirac notation and Hilbert spaces

• 4.2. Inner product of state vectors

• 4.3. Matrix representation of ket and bra vectors

• 4.4. A two-dimensional Hilbert space

and at the end of the lecture there is a set of exercises

• 4.5. Problems

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The contents of this lecture are summarised in the following videos:

• 1. Dirac notation and properties of Hilbert spaces

• 2. Algebra with Dirac notation - bras and kets

• 3. Finding expansion coefficients for Dirac notation

Total lenght of the videos: ~14 minutes

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4.1. Dirac notation and Hilbert spaces

In the previous lecture, we reviewed the basic properties of linear vector spaces. Next, we will discuss how the same formalism can be applied to describe physical states in quantum mechanics.

The state of a physical system in quantum mechanics is represented by a vector belonging to a complex vector space. This vector space is known as the state space of the system.

Ket

!!! info "Ket" A physical state of a quantum system is represented by a symbol |~~\rangle known as a ket. This notation is known as the Dirac notation, and it is very prominent in the description of quantum mechanics. Note that a ket is also refered to as a state vector, ket vector, or just a state.

Hilbert space

The set of all possible state vectors describing a given physical system forms a complex vector space \mathcal{H}, which is known as the Hilbert space of the system. You can think of the Hilbert space as the space populated by all possible states that a quantum system can be found on. Hilbert spaces inherit a number of the important properties of general vector spaces:

!!! info "Superposition" A linear combination (or superposition) of two or more state vectors |{\psi_1}\rangle, |{\psi_2}\rangle, |{\psi_3}\rangle,... |{\psi_n}\rangle, is also a state of the quantum system. Therefore, a linear combination |{\Psi}\rangle of the form |{\Psi}\rangle=c_1|{\psi_1}\rangle+c_2|{\psi_1}\rangle+c_3|{\psi_3}\rangle+...+c_n|{\psi_n}\rangle = \sum_{i=1}^n c_i|{\psi_i}\rangle where c_1, c_2, c_3, ... are general complex numbers will also be a physically allowed state vector of the quantum system.

!!! info "Normalisation" If a physical state of the system is given by a vector |{\Psi}\rangle, then the same physical state can also be represented by the vector c|{\Psi}\rangle where c is a non-zero complex number. The reason for this is that the overall normalisation of the state vector does not change the physics of the system (or in other words, does not modify the information content of the state vector). As we will discuss below, in quantum mechanics it is advantageous to work with normalised vectors, that is, whose length is one. We will define in a while what do we mean by length.