Hilbert space \(\mathcal{H}\): a function space obeying all rules of a linear vector space plus a well-defined scalar product.
Its functions are defined everywhere, square integrable, continuous and infinitely differentiable.
Infinite-dimensional Hilbert space is identified as the \(L^2\) space; \(L^2\) = square-summable sequences: \(\sum_n |x_n|^2<\infty\).
Convergence of \(\sum_n c_n\phi_n\) is ensured via Cauchy sequences — fundamental to the definition of \(\mathcal{H}\).
Elements: vectors/functions \(\phi,\xi,\psi,\ldots\) (Greek letters) and scalars \(a,b,c,\ldots\)
\(\phi,\xi,\psi\) are complex; their conjugates \(\phi^*,\xi^*,\psi^*\) belong to the dual space of \(\mathcal{H}\).
In QM, \(\phi,\xi,\psi\) are states of the system — functions of dynamical variables (generalized coordinates).
Scalars \(a,b,c\) are complex numbers.
A. Addition Rule
Closure: \(\phi(x),\psi(x)\in\mathcal{H}\Rightarrow\phi(x)+\psi(x)=\xi(x)\in\mathcal{H}\).
Commutation: \(\phi(x)+\psi(x)=\psi(x)+\phi(x)\).
Associative: \(\big(\phi(x)+\psi(x)\big)+\xi(x)=\phi(x)+\big(\psi(x)+\xi(x)\big)\).
Null vector: \(O\) exists with \(\psi(x)+O=\psi(x)\).
Inverse: \(-\psi(x)\) exists with \(\psi(x)+(-\psi(x))=O\).
B. Scalar Multiplication
Any linear combination \(a\phi+b\psi\) is also in the space.
Distributive: \((a+b)\psi=a\psi+b\psi\), \(a(\psi+\phi)=a\psi+a\phi\).
Associative: \(a(b\psi)=ab\,\psi\).
Unit and zero scalars: \(I\psi=\psi\), \(0\,\psi=O\).
Scalar Product, Square Integrability and Orthonormality
States are functions of dynamical variables, e.g. position \(\vec r\) and time \(t\).
Scalar product: \((\psi,\phi)=\displaystyle\int\psi^*(\vec r,t)\,\phi(\vec r,t)\,d^3r<\infty\) (\(\psi^*\) in the dual space).
Square integrable: \(\displaystyle\int\psi^*(\vec r)\,\psi(\vec r)\,d^3r<\infty\).
Normalized: \(\displaystyle\int\psi^*(\vec r)\,\psi(\vec r)\,d^3r=1\) (any square-integrable function can be normalized).
Norm: \(N=\sqrt{(\psi,\psi)}\); normalization constant \(A=1/N\).
Orthogonal: \(\displaystyle\int\psi^*(\vec r)\,\phi(\vec r)\,d^3r=0\).
Orthonormal: normalized + orthogonal, \((\phi_i,\phi_j)=\delta_{ij}\) (Kronecker delta).
Example 1.1 Orthonormality
Given \(\phi(x)=\begin{cases}a,&-1\le x\le1\\0,&\text{otherwise}\end{cases}\) and \(\psi(x)=\begin{cases}bx+c,&-1\le x\le1\\0,&\text{otherwise}\end{cases}\). Find \(a,b,c\) (real) such that \(\phi,\psi\) are orthonormal.
Solution
\(\phi\) normalized: \(\displaystyle\int_{-1}^{1}a^2\,dx=1\Rightarrow a=\frac{1}{\sqrt2}\).
Orthogonal: \(\displaystyle\int_{-1}^{1}\frac{1}{\sqrt2}(bx+c)\,dx=0\Rightarrow c=0\).
\(\psi\) normalized: \(\displaystyle\int_{-1}^{1}(bx)^2\,dx=1\Rightarrow b=\sqrt{\tfrac32}\).
Result: \(\phi(x)=\dfrac{1}{\sqrt2}\), \(\psi(x)=\sqrt{\tfrac32}\,x\) on \([-1,1]\) (zero otherwise).
Example 1.2 Norms, normalization constant and orthogonality
\(\phi(x)=e^{-\alpha x^2}\), \(\psi(x)=x\,e^{-\alpha x^2}\), \(-\infty<x<\infty\).
Find the norms of \(\phi,\psi\) and write the normalized functions.
Prove \(\phi\) and \(\psi\) are orthogonal.
Plot \(\phi(x)\) and \(\psi(x)\).
Use \(\displaystyle\int_0^\infty x^n e^{-\beta x^2}dx=\frac12\,\frac{1}{\beta^{(n+1)/2}}\,\Gamma\!\left(\frac{n+1}{2}\right)\).
Solution
(a) \((\phi,\phi)=\displaystyle\int_{-\infty}^{\infty}e^{-2\alpha x^2}dx=\left(\frac{1}{2\alpha}\right)^{1/2}\sqrt\pi\) \(\Rightarrow\; N_1=\left(\dfrac{\pi}{2\alpha}\right)^{1/4}\), \(A_1=\left(\dfrac{2\alpha}{\pi}\right)^{1/4}\).
Normalized: \(\phi(x)=\left(\dfrac{2\alpha}{\pi}\right)^{1/4}e^{-\alpha x^2}\).
\((\psi,\psi)=\displaystyle\int_{-\infty}^{\infty}x^2e^{-2\alpha x^2}dx=\left(\frac{1}{2\alpha}\right)^{3/2}\frac{\sqrt\pi}{2}\) \(\Rightarrow\; N_2=\left(\dfrac12\right)^{1/2}\dfrac{\pi^{1/4}}{(2\alpha)^{3/4}}\)
