Problem 19 - A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

Question

For every bounded operator \(A\) on a Hilbert space \(\mathcal{H}\) show that the exponential operator $$ \mathrm{e}^{A}=\sum_{n=0}^{\infty} \frac{A^{n}}{n !} $$ is well-defined and bounded on \(\mathcal{H}\). Show that (a) \(e^{0}=1\) (b) For all posituve integers \(n,\left(c^{A}\right)^{n}=e^{n A}\). (c) \(\mathrm{e}^{A}\) is invertuble for all bounded operators \(A\) (even if \(A\) is not mivertible) and \(e^{-A}=\left(e^{4}\right)^{-1} .\) (d) If \(A\) and \(B\) are commuting operators then \(e^{A+B}=\mathrm{e}^{A} \mathrm{e}^{\theta}\) (c) If \(A\) is hermitian then \(e^{i A}\) is unitary.

Step-by-Step Solution

Answer

The operator \( e^{A}\) is well-defined and bounded, it has the properties: a) \(e^{0}=1\), b) \((e^{A})^{n} = e^{nA}\) for all positive integers, c) \(e^{A}\) is invertible with \( e^{-A}\) as its inverse, d) If A and B commute, \(e^{A+B} = e^{A}e^{B}\), e) If A is Hermitian, then \( e^{iA}\) is unitary.

Step 1: Proof of bounded operator

This is a straightforward assertion from the definition of operator norm. We first observe that for every \(n\), there is a bounded operator \(A^n\) with \(\|A^n\| \leq \|A\|^n\). The exponential operator serie is therefore absolutely convergent, meaning the operator is well-defined and also norm-bounded.

Step 2: Proof for \(e^{0}=1\)

This part takes advantage from the power series representation of exponential function. When replacing \(A\) by \(0\) in the formula: \[\mathrm{e}^{A}=\sum_{n=0}^{\infty}\frac{A^n}{n!}\]It can be seen that all the terms vanish except when \(n=0\) (where A's exponent equals 0). Therefore, \(e^{0} = 1\)

Step 3: Proof of identity for positive integers

This part can be proved using mathematical induction. For \(n=1\), the statement holds trivially. Assume that the statement holds for some \(n=k\). Then for \(n=k+1\), applying the assumed statement, \( (e^{A})^{k+1} = e^{kA}e^A = e^{(k+1)A}\). Therefore if the statement holds true for \(n=k\), it also holds for \(n=k+1\). By principle of mathematical induction, the statement holds for all positive integers \(n\).

Step 4: Proof of invertibility

Because the exponential operator series is absolutely convergent, we can multiply it term by term with the series of inverse exponential operator \( e^{-A}\). Because \[e^Ae^{-A}=\sum_{n=0}^{\infty}\frac{A^n}{n!}\sum_{m=0}^{\infty}\frac{(-A)^m}{m!} = 1\], it's clear that \( e^A\) is an invertible operator and its inverse is \( e^{-A}\)

Step 5: Proof of commutativity for exponential operators

To prove this, the identity \(AB = BA\) is employed (which is given in the problem statement). Then we can combine \(e^{A+B}\) into a single operator: \(e^{A}e^B\), which is equal to \(e^{B}e^A\) because of the commutativity property.

Step 6: Proof of unitarity for Hermitian operators

\( e^{iA}\) is a unitary operator if it still preserves the inner product: \[ = \] for all \(x, y\) in the Hilbert space. This statement is true due to the condition that A is hermitian.

Key Concepts

Bounded Operator

In the context of a Hilbert space, a "bounded operator" refers to a linear transformation from the space into itself that remains limited in terms of its output magnitude. This concept is essential because it ensures that operators do not lead to infinite values, maintaining mathematical operations well-behaved.

A bounded operator, when applied to any vector in the Hilbert space, results in another vector whose length is proportionate to the initial vector by a fixed constant. This proportionality constant is called the operator norm.

In symbols, a linear operator \(A\) is bounded if there exists a constant \(M\) such that for all vectors \(v\) in the Hilbert space, \(\|Av\| \le M\|v\|\).

The operator norm is defined as \(\|A\| = \sup_{\|v\| = 1} \|Av\|\). It's similar to the concept of the greatest slope or steepness when graphing a straight line on a plane. This property of bounded operators ensures that series like the exponential operator, which is built from adding an infinite number of terms, converges reliably.

Exponential Operator

The exponential operator \( e^A \) is a special operator used frequently in quantum mechanics and other mathematical applications. It extends the familiar exponential function to operators.

The exponential operator is defined by the series: \( e^A = \sum_{n=0}^{\infty} \frac{A^n}{n!} \). This representation connects it to the ordinary exponential function \( e^x \) where \(x\) is a real number.

• It uses an infinite series to approximate \(e^A\), encompassing all integral powers of \(A\).
• The convergence and bounded nature of the series are assured when \(A\) itself is bounded.

This series converges absolutely thanks to the bounded nature of \(A\), meaning the sum of the series is not affected by changing the sequence of terms. Consequently, the exponential operator retains the boundedness, ensuring stability within the Hilbert space.

Commuting Operators

Commuting operators are a fundamental part of mathematical physics, particularly quantum mechanics, where operations often depend on their order. Two operators \(A\) and \(B\) are said to commute if applying them in sequence yields the same result irrespective of the order: \(AB = BA\).

When dealing with exponential operators, this property plays a crucial role. For two commuting operators \(A\) and \(B\), the exponential operator satisfies \( e^{A+B} = e^A e^B \).

Here are key features of commuting operators:

• Their commutativity ensures predictable results. Order of operations does not alter outcomes.
• This property is instrumental in simplifying complex calculations in quantum mechanics and other advanced mathematical contexts.

If operators do not commute, different techniques are required as the lack of commutativity makes relational mathematical expressions much more complex.

Hermitian Operator

A Hermitian operator, also known as a self-adjoint operator, is crucial in quantum mechanics because its eigenvalues represent measurable quantities and are always real numbers.

For an operator \(A\) to be Hermitian, it must satisfy the condition: \( \langle Av, w \rangle = \langle v, Aw \rangle \) for all vectors \(v,w\) in a Hilbert space.

Hermitian operators take center stage when applying the exponential operator in contexts involving complex arithmetic, such as \( e^{iA} \), which represents unitary operations. These preserve the inner product, ensuring no distortion in length or angle of vectors in the space:

• A unitary operator \( e^{iA} \) maintains \( \langle e^{iA}x, e^{iA}y \rangle = \langle x, y \rangle \).
• The Hermitian property of \(A\) suggests that transformations are rotation-like in nature.

Hermitian operators also guarantee the resulting exponential operator is unitary, meaning it corresponds to an energy-conserving operation that fits seamlessly within the framework of quantum mechanics.