Given a ordinary function f(x) and an operator R, the...

Given a ordinary function $f(x)$ and an operator $R$, the operator $f(R)$ is defined to be $$ f(R)=\sum_{i} f\left(r_{i}\right)\left|r_{i}\right\rangle\left\langle r_{i}\right| $$ where $r_{i}$ are the eigenvalues of $R$ and $\left|r_{i}\right\rangle$ are the associated eigenkets. Show that when $f(x)=x^{2}$ this definition implies that $f(R)=R R$, that is, that operating with $f(R)$ is equivalent to applying the operator $R$ twice. What bearing does this result have in the meaning of $\mathrm{e}^{R} ?$

Given a ordinary function $f(x)$ and an operator $R$, the operator $f(R)$ is defined to be
$$
f(R)=\sum_{i} f\left(r_{i}\right)\left|r_{i}\right\rangle\left\langle r_{i}\right|
$$
where $r_{i}$ are the eigenvalues of $R$ and $\left|r_{i}\right\rangle$ are the associated eigenkets. Show that when $f(x)=x^{2}$ this definition implies that $f(R)=R R$, that is, that operating with $f(R)$ is equivalent to applying the operator $R$ twice. What bearing does this result have in the meaning of $\mathrm{e}^{R} ?$

Key Concepts

Spectral Decomposition

Spectral decomposition is a method in linear algebra and quantum mechanics where an operator is expressed in terms of its eigenvalues and their corresponding eigenvectors (or eigenkets). This representation is central to understanding the structure of linear operators, allowing one to analyze and manipulate the operator in a basis where it has a simple, often diagonal, form.

Functional Calculus

Functional calculus refers to the process of defining and computing functions of operators by applying the function directly to the eigenvalues in the spectral decomposition. This approach allows one to extend familiar functions from numbers to operators, maintaining meaningful analogies such as f(R) operating on eigenstates with eigenvalues f(r).

Operator Multiplication

Operator multiplication in the context of the spectral theorem means that the square of an operator (or more generally any power) can be computed by multiplying its eigenvalue expression accordingly. By applying a function like f(x)=x^2 via the spectral theorem, one finds that f(R)=R^2, because applying R twice is equivalent to squaring each eigenvalue.

Exponential of an Operator

The exponential of an operator, defined by its power series expansion, is a natural extension of the exponential function to the operator setting. Through functional calculus, e^R is obtained by applying the exponential function to each eigenvalue, resulting in e^R acting on an eigenvector with eigenvalue r as multiplication by e^r. This ties the concept of exponentiation directly to spectral decomposition, making it a key tool in solving differential equations and describing time evolution in quantum mechanics.

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