Quantum Formalism: Vectors
Linear Algebra: a Boring Prelude to Quantum Mechanics
The purpose of this article is to develop the mathematical formalism of quantum mechanics. Terminology, notation, and conceptual background form the groundwork of this theory. Needless to say, this isn’t the good stuff yet but you need to survive linear algebra to get somewhere. Why is this important? Well, the state of a physical system in quantum mechanics is represented by a vector, belonging to a complex vector space. This vector is known as the state space of the system. We’ll get to that in a minute. I trust you are familiar with the idea of what a vector is to begin with — they are quantities with both, magnitude and direction.
A quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system’s evolution in time exhausts all that can be predicted about the system’s behavior. A mixture of quantum states is again a quantum state. A mixture of a mixture of quantum states is still a quantum state. You get the point. Quantum states that cannot be written as a mixture of other states are called pure quantum states, while all other states are called mixed quantum states. Pure states are also known as state vectors or wave functions, the latter term applying particularly when they are represented as functions of position or momentum, Ψ.
Such a physical state of a quantum system is represented by a symbol | ⟩, known as a ket. This notation is known as the Dirac notation, and it is very prominent in the description of quantum mechanics. More importantly, the set of all possible states describing a given physical system forms a complex vector space H, which is known as the Hilbert space of the system. The idea of Hilbert spaces is central to quantum formalisms. It’s extremely difficult for me to explain what Hilbert spaces are in an article that is not about Hilbert spaces so we will leave the details for another day. You can think of the Hilbert space as the space populated by all possible states that a quantum system can be found in. These are infinite dimensional spaces that are merely mathematical in the sense that they are physically insignificant.
A Hilbert space is a type of vector space.
More generally, a vector space consists of a set of vectors that define all possible quantum states, {|α⟩, |β⟩, |γ⟩, …} and a set of scalars that appropriately scale the vectors, {a, b, c, …}. Vector spaces are subjected to many defining properties. A heads up: I will be stating the obvious.
Vector arithmetic
Quantum states work exactly as you would expect them to. The arithmetic governing them and vectors is the same. Here are some fundamental operations.
Quantum state addition: the sum of any two states is another state
Addition is commutative,
and associative
There also exists a null state, |0⟩, with the property that
For every state, there is an associated inverse state such that
Scalar multiplication
The product of any scalar with a vector is another vector:
Scalar multiplication is distributive:
It is also associative
I seriously recommend reading this if you are not familiar with eigenstuffs. It will certainly help motivate this article.
So far, I have merely stated the obvious. There is a lot less here than meets the eye. All I have done is write down abstract vector arithmetic in the context of quantum states. But here is where things get interesting. We introduce an idea called linear independence. Given a certain vector space, in arbitrary dimensions, we can specify the number of independent directions. This means that for two vector spaces with the same dimension, the properties that depend only on the vector-space structure are exactly the same (we say that they are isomorphic). A vector space is finite-dimensional if its dimension is a natural number. Otherwise, it is infinite-dimensional, and its dimension is an infinite cardinal. Finite-dimensional vector spaces occur naturally in most physics. Infinite-dimensional vector spaces arise in many areas of mathematics and here as well. We, however, do not care much for the maths. The physicist is always interested in the special case.
Usually, we denote vectors as a letter with an arrow on top. Any physical vector belongs to a certain finite dimensional vector space. In quantum mechanics, however, we deal with infinite dimensional vector spaces. To denote this, we employ the bra-ket notation. As we’ve already demonstrated, kets behave exactly as vectors do. You can add them, multiply them, divide them, and generalize them. To introduce the quantum analog of multiplying two vectors, you’d need to invoke the bra vector.
The bra vector is the Hermitian conjugate of a ket. What does this mean? Well, the Hermitian conjugate can also be thought of as the conjugate transpose of a matrix. The transpose of a matrix is obtained by changing the rows into columns and columns into rows for a given matrix. Note that the diagonal remains the same even with our transpose.
Quantum vectors often take complex-valued arguments. Each ket vector in some Hilbert space, H, has an associated bra vector in a dual space H*. Here is how that works.
The bra takes the opposite notation of the ket. Instead of |⟩, we work with ⟨|.
The “bra” is similar, but the values are the transpose of the ket, and each element is the complex conjugate of the same.
And multiplying a bra with a ket looks something like this, ⟨v|w⟩.
So, if a ket vector is given by |Ψ⟩= a|v⟩ + b|w⟩, the corresponding bra vector is ⟨Ψ|= a*⟨v| + b*⟨w|. Where a* and b* are complex conjugates of a and b.
As mentioned above, the vector space spanned by all bra vectors ⟨Ψ| is referred to as the dual space and is represented by H*. For each ket vector belonging to H, there will exist an associated bra vector belonging to the dual space H*.
Basis Vectors and Linear Independence
If a vector, |λ⟩, cannot be written as a linear combination of the vectors defining our vector space, then it is known to be linearly independent. Almost like how a unit vector in the z direction is linearly independent of the unit vectors in the x and y direction but any vector on the x-y plane can be represented as a linear combination of the x and y unit vectors. By extension, a set of vectors are said to span a vector space if and only if every vector can be written as a linear combination of this set.
A set of vectors that spans a space is called a basis and the number of vectors in the basis is called the dimension of the vector space.
As a result, with respect to an established basis,
any given vector can be considered an element of the set that defines the vector space. Therefore, a linear combination of two or more state vectors,|ϕ⟩, is also a state of the quantum system. We can then say that a linear combination, |Ψ⟩, of the form
where c1, c2, c3,… are general complex numbers will also be a physically allowed state vector of the system.
Another property worth noting is that if a physical state of a quantum system is given by a state vector,|Ψ⟩, then the same physical state can also be represented by the vector c|Ψ⟩ where c is a non-zero complex number. The reason for this is that the overall normalization of the state vector does not change the function of the system. In other words, scaling the state vector does not modify the information content of the system. In quantum mechanics, it is advantageous to work with normalized vectors, that is, whose length is one. However, normalization is merely a mathematical tool we employ to make it fit our theories. Probability can never be greater than one and since the state vector is directly related to the probability distribution of the system, it is incredibly important for it to be normalized. But the fact that a quantum system is not normalized does not affect the physical information content that we can exhaust from the system.
We can then establish that a set of vectors is complete if every state of the quantum system can be represented as a linear combination of them. In such a case, it becomes possible to express any state vector |Ψ⟩ of the system’s Hilbert space as a superposition of these n vectors. The set of vectors {|ϕ⟩} is then said to span the Hilbert space of the quantum system.
What we’ve demonstrated here is the classical equivalent of the idea that if every vector in a vector space can be defined by a linear superposition of the bases, then these vectors span the complete quantum system.
So, a set of vectors {|ϕi⟩} is said to form a basis for the state space if the set of vectors is complete and if in addition, they are linearly independent. The latter condition means essentially that one cannot express a given basis vector as a linear combination of the rest of the basis vectors. Linear independence can also be expressed as the following requirement
The minimum number of vectors needed to form a complete set of basis states is known as the dimensionality of the state space. In quantum mechanisms, you will probably encounter systems whose Hilbert spaces have very different dimensionality. It becomes incredibly important to obtain the complete picture of the systems you’re working with.
With that, I will end this here. I hope this has provided some background to the formalisms of Hilbert spaces and quantum vectors. Thank you for reading!
