Space and momentum representation

Part 1: see script of Vorlesung (November) Lutz Polley part.


Part 2: The (idealized) reprentations of states of physical systems by using either space or momentum dependent functions are often useful in practice, good for the imagination and easy and mnemonic to handle. To get used to the first time and gain some practise is the aim of this chapter.


The counters To gain a representation of a physical state, -say a particle hanging around,say in a state $\vert\psi\!>$, needs a complete and disjunct set of counters to detect it experimentally. We will restrict this chapter to one space coordinate, no fuss to translate it to more space dimensions for the reader.


Let $\vert x_i,\Delta x\!>$ be such a counter at position i and with a width of $\Delta x$. Then in putting them online in a row on the x-axis, touching each other but not overlapping, to gain complete information on the 'where is the particle' experimental question, the answer will come as follows: for a single experiment one of the (countable infinite many) counters will say 'yes', the others all will say 'no, not here'. Not much to learn from this. However, in repeating the same experimental setup and initial state many times the statistics of individual counts comes to a unique answer, giving the full information on the state of the particle before measurement, namely the prepared state $\vert\psi\!>$. Weomit in the notation the clumsy fixed $\Delta x$ but keep it in mind.


Formally we have the representation
 $$\begin{align}\vert\psi\!> & = \sum_{n=1}^\infty \psi_n \vert x_n\!> \\ W(x_n) & = \arrowvert \psi_n * \psi_n \arrowvert \end{align}$$
and all of the counters form a complete basis set of states for the Hilbert-space of states of the particle before measurement.
$$\begin{align}\{ \vert x_i\!> \} \quad \forall i=1,2,3,\ldots \end{align}$$
and by forming the scalar product of $\vert\psi\!>$ with any x_i we understand the formal writing of the coefficients in [#!l2-1!#],

$$<\!x_i\vert\psi\!> = \sum_n \psi_n <\!x_i\vert x_n\!> = \sum_n \psi_n \delta_{i,n} = \psi_i \quad .$$ (3)

The more elegant notation for the disjunct but complete set of counters are the completeness and the orthonormality
$$\begin{align}\sum_n \vert x_n\!><\!x_n\vert & = 1 \quad , \\ <\!x_n\vert x_m\!> & = \delta_{nm} \quad . \\ \end{align}$$
In the ordinar space-vector space we have analogously
$$\begin{align}( \vec{e_i} \arrowvert \vec{e_j} ) &= \delta_{ij} \\ \sum_{i=1}^3 \vec{e_i} \circ \vec{e_j} &=1\\ \end{align}$$
where the 'dyadic product $\vec{e_i} \circ \vec{e_j}$applied to any vector $\vec{a}$ gives

$$\sum_i \vec{e_i} \circ \vec{e_i} \quad \vec{a} = \sum_i \vec{... ...c{a} ) = \sum_i \vec{e_i} a_i = \sum_i a_i \vec{e_i} = \vec{a}$$ (4)