Basics Quantum Mechanics

Last updated Dec 14, 2022

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In QM the state of a closed system is fully described by a state vektor $\ket{\psi}$ in a Hilbert space $\mathcal{H}$. The Dual space $\mathcal{H}^*$ contains all Linear maps $\bra{\phi}$ from the Hilbert space to the complex numbers.

The scalar product of the Hilbert space is $\braket{\phi|\psi}$ and the norm $||\psi||=\sqrt{\braket{\psi|\psi}}$.

The space is seperatable if there is a countable basis $\ket{\phi_a}$ with $\braket{\phi_a|\phi_b}=\delta_{ab}$, so that for every vector $\ket{\psi}$ there exists the decomposition $$\ket{\psi} = \sum_a c_a\ket{\phi_a},$$ with the coefficients $c_a = \braket{\phi_a|\psi}$.

Observables are represented by self adjoint and linear operators $\hat{R}$. Self adjoint means $\hat{R}\ket{\psi} = \hat{R}^\dagger\ket{\psi}$.

The general definition of self adjoint or hermitian is through the scalar product $<.,.>$ of the respective space $$<Ax,y> = <x,Ay> \forall x,y,$$ which is equivalent to the notation $\braket{x|A|y} = \braket{x|A^\dagger|y}$. With $x=y$ this leads to the fact that the Eigenvalues of a hermitian operator are real valued.

If the operator $A$ is hermitian on a space $V$ there is an orthonormal basis of $V$ given by eigenvektors of $A$ The operator $A$ can be written as a linear combination of the projection operators on its eigenvectors: $$A\ket{\psi_n} = \lambda_n \ket{\psi_n} \Rightarrow A = \sum_n \lambda_n \ket{\psi_n}\bra{\psi_n},$$ which follows directly from the completeness relation of the basis $$\sum_n \ket{\psi_n}\bra{\psi_n} = 1.$$

Another formulation of the theorem is as integral with a projection-valued measure $$A = \int \lambda dE_\lambda,$$ where $E_\lambda$ is a parametrized family of commuting orthogonal projection operators (Same formulation in continous case ?). It satisfies $\lim_{\lambda\to -\infty}E_\lambda = 0$ and $\lim_{\lambda\to \infty}E_\lambda = 1$ The definition can be extended to arbitrary functions $f$ $$f(A) = \int f(\lambda) dE_\lambda.$$

To define projection and completenss relations we devide the space into parts of the parameter $\Delta E_{\lambda_\alpha} = E_{\lambda_\alpha} - E_{\lambda_{\alpha-1}}$ which leads to the completess relation $$\sum_\alpha \Delta E_{\lambda_\alpha}.$$

The spectrum of the operator can be decomposed into a discrete and a continous part The discrete spectrum contains jump points which are not continous from the left in the parameter $\lambda$ $$\exists \epsilon: E_\lambda - E_{\lambda-\epsilon} \neq 0.$$ The continous spectrum si continous in $\lambda$

Example: Discrete spectrum As formulated above the easiest example is an operator with a discrete spectrum (like matrix) For $$\hat{R}\ket{\psi_{n,k}} = \lambda_{n}\ket{\psi_{n,k}},$$ where $n$ labels the different eigenvalues and $k$ the different eigenvectors for the same eigenvalue.

The projection operators on the space for a single eigenvalue are just $$\Pi_n = \sum_k \ket{\psi_{n,k}}\bra{\psi_{n,k}},$$ from which one can construct the spectral family $E_r$: $$E_r = \sum_{r_n \leq r} \Pi_n$$ This way we obtain the usual decomposition of the operator in terms of the basis projectors $$\hat{R} = \int_{-\infty}^{\infty} r dE_r = \sum_n r_n \Pi_n$$

Example: position operator $\hat{x}$

In position space the position operator $\hat{x}$ gives $$\hat{x}\psi(x) = x\psi(x),$$ and thus has a continous spectrum The spectral family $E_r$ becomes $$E_r \psi(x) = \Theta(r - x)\psi(x)$$