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
$$\psi(L,t)=⟨\phi|\psi⟩=\int_{-∞}^{∞} \phi^*{\psi}\text{ dL}.\tag{8}$$