Quantum Mechanics (1) · Problem 6

A few things first

Physics — Quantum Mechanics · operators-infinite-dimensional-spaces

Problem

Consider the following equation in

E_\infty

(infinite-dimensional Euclidean space -- let the scalar product be

\langle x|y\rangle \equiv \sum_{n=1}^\infty x_n^* y_n

Cx = a,

where the operator

C

is defined by (in some basis):

C(x_1, x_2, \ldots) = (0, x_1, x_2, \ldots)

C

: (a) A bounded operator [i.e., does there exist a non-negative real number

\alpha

such that, for every

x \in E_\infty

, we have

|Cx| \leq \alpha|x|

(

"|x|"

denotes the norm:

\sqrt{\langle x|x\rangle}

)]? (b) A linear operator? (c) A hermitian operator (i.e., does

\langle x|Cy\rangle = \langle Cx|y\rangle

)? (d) Does

Cx = 0

have a non-trivial solution? Does

Cx = a

always have a solution? Now answer the same questions for the operator defined by:

G(\alpha_1, \alpha_2, \ldots) = (\alpha_1, \alpha_2/2, \alpha_3/3, \ldots). \tag{1.127}

Note that we require a vector to be normalizable if it is to belong to

E_\infty

-- i.e., the scalar product of a vector with itself must exist.