Quantum Mechanics 101: The Math 🧮
In school, you’ve probably learned about classical mechanics — centripetal force, acceleration, kinematic equations, torque, and more. Quantum mechanics is much different; it studies subatomic particles 🧪. There are many cool properties and cool ideas behind quantum mechanics, and quantum mechanics has a lot of applications in the world of quantum cryptography and quantum computing 💻.
You may think that quantum mechanics is heavily focused on theory, where much math is required. After all, it would be really hard to calculate in the subatomic level? Well, behind quantum mechanics is a lot of math 🧮. Specifically, there is a lot of linear algebra. Do not worry if the math is intimidating! It does take time to understand and comprehend. As a high school student myself, I tried my best to make the math sound as less intimidating as possible.
Linear Algebra
Behind quantum mechanics is linear algebra. Linear algebra allows for the effective representation of quantum operators and quantum states. It is essentially the language of quantum mechanics. Once you are comfortable with linear algebra, it will be a lot easier to understand quantum mechanics and a lot of the terminology used in quantum mechanics.
Vectors
Vectors are extremely important in quantum mechanics. Vectors essentially allow you to stack numbers in a column or a row.
You can do many operations with vectors. You can add them together. You could also multiply a vector with a number, often called a scalar.
A vector space, which will be really important later on when we define key terminology, is essentially a set of mathematical objects that are able to do scalar multiplication and element-wise addition.
Vectors can also be represented as arrows. Two vectors are orthogonal when they are perpendicular to each other. We will understand the mathematical implications of this when we talk about inner products.
Matrices
Matrices are essentially a grid of numbers. In a similar fashion to vectors, you can add matrices and perform scalar multiplication as well. Matrices can be thought of as a stack of multiple vectors.
There are two types of special matrices: identity matrices and unitary matrices. These matrices are special because they have special properties are can be useful at times, especially in quantum mechanics.
Aside from identity matrices, unitary matrices are also extremely important. We’ll talk more about unitary matrices shortly.
Conjugate and Transpose Operations
The conjugate of a matrix is essentially taking a complex conjugate of all the numbers within the matrix. A complex conjugate involves imaginary numbers. A complex conjugate of a complex number is essentially keeping the sign of a real number but inverting the sign of the complex number.
The transpose operation is essentially an operation that flips a matrix diagonally.
When doing both operations at the same time, it is called a conjugate transpose.
A unitary matrix is essentially when the conjugate transpose of a matrix multiplied by the matrix itself is equal to the identity matrix. These unitary matrices are extremely important in quantum mechanics, especially with quantum operations.
Inner Products
The Hilbert space is essentially a vector space where inner products are also defined. When an inner product between two vectors is 0, then we call them orthogonal. This is important terminology in quantum mechanics, and if you decide to read more papers about quantum mechanics, you will hear the term a lot.
Taking the inner product of a vector also allows us to find the length of the vector, also called the norm of the vector. To normalize a vector, you essentially need to scale the vector by its norm.
Basis
In linear algebra, you can describe any vector using a finite set of vectors, called the basis. However, the basis must be linearly independent, meaning that I cannot represent one of the vectors as another within the basis.
The Tensor Product
The tensor product, also called the Kronecker product, allows us to expand a vector. They are seen a lot in the world of quantum mechanics, especially in quantum computing when representing qubits.
Essentially, in a tensor product, you get each element of the first vector and multiply each element of the first vector by the second vector. This will ultimately expand the vector.
Applying Linear Algebra to Quantum Mechanics
Wave Particle Duality
One of the key notions in quantum mechanics is wave particle duality. Particles can have properties of waves.
For example, constructive and destructive interference are properties of waves. These types of quantum interference are seen a lot in quantum mechanics.
The Braket notation
So far, in linear algebra, we’ve been using a lot of different notation. When we move onto quantum mechanics, we are introduced to braket notation. Why do we use a different set of notation? 🤔 Well, braket notation has a lot of cool and neat properties and patterns, which can be really useful when you study quantum mechanics even further.
Using a ket, you can represent a vector.
Using a bra, you can represent a conjugate transpose of a vector. A bra is essentially an inverted ket.
Using a braket, you can represent the inner product of two vectors.
Quantum States
A quantum state is essentially physical properties of a system. These can be properties such as position, momentum and spin. Connecting this to linear algebra, quantum states can be represented as unit vectors in complex Hilbert space.
A wave function in quantum mechanics is essentially a function that holds the information about a quantum state. This wave function is represented with a psi.
Quantum superposition, meaning that a system can be in two states at once, can be represented through linear algebra.
To measure if a system is in a given state, we can use Born’s rule. The probability of a quantum system being in a space state is essentially taking the absolute value and squaring the inner product between the state and the system, represented through a function called a wave function.
The wave function essentially collapses into a singular state once measured.
Quantum Operations
We can transform a given state to another state using quantum operations. Quantum operations can be represented as unitary matrices on Hilbert space.
The most popular quantum operations can be represented through special matrices called Pauli matrices.
Hermitian operators are essentially matrices that are conjugate transposes of themselves.
Observables
Observables are measurable properties of a quantum system. Observables can be represented as eigenvalues. These observables can be calculated when applying a hermitian operator, which are matrices essentially, on Hilbert space. Eigenvalues and eigenvectors are beyond the space of this article as they can get very complicated.
A famous example of this property is the Schrödinger equation. When applying the Hamiltonian operator, which is a hermitian operator, we can get the energy of a quantum state.
Conclusion
Quantum mechanics is really cool. They have a ton of applications, especially in cryptography. Shor’s algorithm is extremely interesting and it can revolutionize our current methods of cryptography. It is really exciting, and this article just scratches the surface of quantum mechanics and how it works inherently.
Thank you for reading and I hope you learned something new! All feedback is appreciated. If you have any questions or feedback, please email me at p.thanosan23@gmail.com.
