# Fundamental Principles and Postulates of Quantum Mechanics

Principle 1: Whenever you measure any physical quantity $$L$$, there is a Hermitian linear operator $$\hat{L}$$ (called an observable) associated with that measurement.

Principle 2: Any arbitrary state of a quantum system is represented by a ket vector $$|\psi⟩$$.

Principle 3: The possible measurable values of any quantity are the eigenvalues $$λ_L(=L)$$ of $$\hat{L}$$.

Principle 4: According to the Copenhagen interpretation of quantum mechanics, after measuring $$L$$, the possible states a quantum system can end up in are the eigenvectors $$|λ_L⟩(=|L⟩$$) of $$\hat{L}$$.

Principle 5: For any two states $$|\psi⟩$$ and $$|\phi⟩$$, the probability amplitude of the state changing from $$|\psi⟩$$ to $$|\phi⟩$$ is given by

$$\psi=⟨\phi|\psi⟩.\tag{1}$$

The probability $$P$$ of the state changing from $$|\psi⟩$$ and $$|\phi⟩$$ can be calculated from the probability amplitude using the relationship

$$P=\psi^*\psi=⟨\psi|\phi⟩^*⟨\phi|\psi⟩=|\psi|^2.\tag{2}$$

From a purely mathematical point of view, any ket $$|\psi⟩$$ in Hilbert space can be represented as a linear combination of basis vectors:

$$|\psi⟩=\sum_i{\psi_i|i⟩}.\tag{3}$$

The kets $$|1⟩\text{, ... ,}|n⟩$$ represent any basis vectors and their coefficients $$\psi_1\text{, ... ,}\psi_n$$ are, in general, complex numbers. We shall prove in the following sections that we can always find eigenvectors $$|L_1⟩\text{, ... ,}|L_n⟩$$ of any observable $$\hat{L}$$ that form a complete set of orthonormal basis vectors; therefore any state vector $$|\psi⟩$$ can be represented as

$$|\psi⟩=\sum_i{\psi_i|L_i⟩}.\tag{4}$$

We’ll also prove that the collection of numbers $$\psi_i$$ are given by

$$\psi_i=⟨L_i|\psi⟩\tag{5}$$

and represent the probability amplitude of a quantum system changing from the state $$|\psi⟩$$ to one of the eigenstates $$|L_i⟩$$ after a measurement of $$L$$ is performed. The collection of all the probability amplitudes $$\psi_i$$ is called the wavefunction. When the wavefunction $$\psi(L,t)$$ associated with the state $$|\psi⟩$$ becomes a continuous function of $$L$$ (that is, the range of possible values of $$L$$ becomes infinite and the number of probability amplitudes becomes infinite), we define $$|\psi|^2$$ as the probability density. One example where $$\psi$$ becomes continuous is for a particle which can have an infinite number of possible $$x$$ positions. Then $$\psi$$ becomes a continuous function of $$x$$ (and, in general, also time $$t$$). Since $$|\psi(x,t)|^2$$ is the probability density, the product $$|\psi(x,t)|^2dx$$ is the probability of measuring $$L$$ at the position $$x$$ and at the time $$t$$. The probability of measuring anything at the exact location $$x$$ is in general zero. A far more useful question to ask is: what is the probability $$P(x+Δx,t)$$ of measuring $$L$$ within the range of x-values, $$x+Δx$$? This is given by the following equation:
$$P(x+Δx,t)=\int_{x}^{x+Δx} |\psi(x,t)|^2dx.\tag{6}$$
According to the normalization condition, the total probability of measuring $$L$$ over all possible values of $$x$$ must satisfy
$$P(x,t)=\int_{-∞}^{∞} |\psi(x,t)|^2dx=1.\tag{7}$$
If $$\psi(L,t)$$ is continuous, then the inner product $$⟨\phi|\psi⟩$$ is defined as
$$\psi(L,t)=⟨\phi|\psi⟩=\int_{-∞}^{∞} \phi^*{\psi}\text{ dL}.\tag{8}$$