Vectors and Inner Products in Quantum Mechanics
Generalization of Vectors
For science students who have not learned linear algebra, a vector is a physical quantity with magnitude and direction, meaning a point in 3-dimensional space, and it is usually represented as in $\vec{x} = (x_{1}, x_{2}, x_{3})$. This definition poses no significant problem in studying classical mechanics and electromagnetism. However, in quantum mechanics, concepts like Fourier analysis and inner product of functions arise, so if one does not know the generalized definition of vectors, they may face great difficulty in their studies.
In linear algebra, a vector is an abstraction of what we intuitively think of as a vector. All things that share the same properties as a vector in 3-dimensional space are called vectors, and the set of these vectors is called a vector space. These properties are those that we naturally expect a point in 3-dimensional space to satisfy. For example:
- The sum of two vectors is also a vector.
- Multiplying a vector by a scalar results in a vector.
Accordingly, points in 3-dimensional space are vectors, and the 3-dimensional space itself becomes a vector space. Below are two examples most crucial in quantum mechanics. Matrices and functions are also vectors.
Examples
Matrices
Consider the set of matrices with size $m \times n$. Even if you add these matrices, the result is still a $m \times n$ matrix, and multiplying any matrix by a scalar also results in a $m \times n$ matrix, hence this set becomes a vector space, and each matrix becomes a vector.
If one recalls that writing expressions such as $\mathbf{x} = (x_{1}, x_{2}, x_{3})$ in ordered pairs or representing them as $1 \times 3$ matrices results in no substantial differences, the idea that matrices are vectors becomes more intuitive.
Functions
Consider the set of continuous functions. If $f$ and $g$ are continuous functions, then their sum $f+g$ is also a continuous function. Additionally, any scalar multiple of $cf$ is still a continuous function. Therefore, the set of continuous functions becomes a vector space, and each continuous function becomes a vector.
In fact, remember that when dealing with vector functions whose function values are 3-dimensional vectors, they can be expressed as follows:
$$ f(x,y,z) = (xy, yz, z^{2}) $$
Generalizing the Inner Product
The inner product is a very useful operation when dealing with vectors. Just as the concept of a vector was generalized, let’s generalize the inner product. First, the notation for the generalized inner product uses angle brackets $\braket{\ ,}$ instead of the dot $\cdot$. If we have $\mathbf{x} = \left( x_{1}, x_{2}, x_{3} \right)$ and $\mathbf{y}=\left( y_{1}, y_{2}, y_{3} \right)$, it is represented as follows:
$$ \mathbf{x} \cdot \mathbf{y} = x_{1}y_{1} + x_{2}y_{2} + x_{3}y_{3} = \braket{\mathbf{x}, \mathbf{y}} $$
In quantum mechanics, a vertical bar $|$ is used instead of a comma in the middle.
$$ \mathbf{x} \cdot \mathbf{y} = \braket{\mathbf{x} \vert \mathbf{y} } $$
This is called Dirac notation. If the inner product of two vectors is $0$, the two vectors are said to be orthogonal and are expressed as follows:
$$ \mathbf{x} \perp \mathbf{y} \quad \iff \quad \braket{\mathbf{x} \vert \mathbf{y}} = 0 $$
The key to generalizing a vector is that ‘whatever satisfies the properties that we think vectors should satisfy’ is called a vector. Similarly, for the inner product, we maintain the concept of ‘summing the product of each component’. Depending on the vector space being considered, the definition of the inner product changes as follows.
Examples
Matrices
Let us consider two matrices $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}, B = \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}$. The inner product of these two is defined as ’the sum of the product of each component’, just like the inner product of 3-dimensional vectors.
$$ \braket{ A \vert B } = a_{11}b_{11} + a_{12}b_{12} + a_{21}b_{21} + a_{22}b_{22} $$
Functions
Given that functions are also vectors, we can define the inner product of two functions. The inner product of functions is defined by the following definite integral.
$$ \braket{\psi \vert \phi} = \int \psi^{\ast}(x) \phi (x) dx $$
Here, $\psi^{\ast}$ denotes the complex conjugate of $\psi$. Be cautious of ambiguities in the notation. For the reason why the inner product of functions is defined as such, refer to ‘Why the inner product of functions is defined as a definite integral’.