State Vectors in Hilbert Space

Mathematically, any physical state of a system is represented by a vector Y ) in a corresponding Hilbert space 7i. In general, a Hilbert space is a linear space (i.e., vector space) of infinite dimension, where the vectors are square-integrable complex-valued functions of space and time. The integrability constraint. 

This concept of direct products — also called Kronecker or tensor products -will be elaborated in detail in section 8.4. 

Formally, any physical observable is then a linear, hermitean operator 0, 

Any statement concerning the physical properties of the system has thus to be expressed in terms of mathematical operations defined on this Hilbert space. 

Upon our transition from classical mechanics to quantum mechanics, we distinguish operators from their classical analogs by a hat. This is convenient here but will make all further derivations rather clumsy. Therefore, the hat will be omitted as a designator for an operator later in this and then in all following chapters. It should be obvious from the context which quantity is an operator and which is not. 

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Because the product (3.8) is a probability, it must integrate to 1.0 when all possible outcomes are taken into account. Consequently, the wave functions are multiplied by arbitrary constants (n ),/2 chosen to make this integral come out to 1.0 over the complete range of motion. These are called normalization constants. It is legitimate to multiply solutions to the Schroedinger equation by an arbitrary constant because they are elements of a closed binary vector space. Multiplication of a solution by any scalar yields another element in the space, hence the product of the normalization constant and the wave function (or any other state vector in Hilbert space) also is a solution. 

In quantum mechanics, dynamical variables are represented by linear Hermitian operators 0 that operate on state vectors in Hilbert space. The spectra of these operators determine possible values of the physical quantities that they represent. Unlike classical systems, specifying the state ) of a quantum system does not necessarily imply exact knowledge of the value of a dynarttical variable. Only for cases in which the system is in an eigenstate of a dynamical variable will the knowledge of that state IV ) provide an exact value. Otherwise, we can only determine the quantum average of the dynamical variable. 

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