Question

Evaluate the following absolute square of a complex number (which arises in a problem in quantum mechanics). Assume \(a\) and \(b\) are real. Express your answer in terms of a hyperbolic function. $$ \left|\frac{(a+b i)^{2} e^{b}-(a-b i)^{2} e^{-b}}{4 a b i e^{-i a}}\right|^{2} $$

Short answer

Simplify the numerator using hyperbolic functions, then divide by the denominator and finally, find the modulus.

Step-by-step solution

Simplify the Numerator

Consider the numerator \((a + b i)^{2} e^{b} - (a - b i)^{2} e^{-b}\). Expand each term separately:\[ (a + b i)^{2} = a^2 + 2abi - b^2 \]\[ (a - b i)^{2} = a^2 - 2abi - b^2 \]Now, substitute these into the expression:\[ (a^2 + 2abi - b^2)e^b - (a^2 - 2abi - b^2)e^{-b} \]

Combine Terms

Combine the terms from the expanded expression using like terms:\[(a^2 + 2abi - b^2)e^b - (a^2 - 2abi - b^2)e^{-b} \]Separate the real and imaginary parts:\[ a^2(e^b - e^{-b}) + 2abi(e^b + e^{-b}) - b^2(e^b - e^{-b}) \]This gives: \[ (a^2 - b^2)(e^b - e^{-b}) + 2abi(e^b + e^{-b}) \]

Simplify the Numerator Further Using Hyperbolic Functions

Recall the hyperbolic sine and cosine definitions:\[ \text{sinh}(x) = \frac{e^x - e^{-x}}{2} \]\[ \text{cosh}(x) = \frac{e^x + e^{-x}}{2} \]Using these, rewrite the terms in the numerator:\[ (a^2 - b^2)(2\text{sinh}(b)) + 2abi(2\text{cosh}(b)) \]Factor out constants:\[ 2(a^2 - b^2)\text{sinh}(b) + 4abi\text{cosh}(b) \]

Simplify the Denominator

Rewrite the denominator: \[ 4abi e^{-ia} \]

Rewrite the Entire Expression

Combine the simplified numerator and denominator:\[ \frac{2(a^2 - b^2)\text{sinh}(b) + 4abi\text{cosh}(b)}{4abi e^{-ia}} \]

Simplify and Normalize the Expression

Simplify the expression by dividing through by the common factor of 2 and simplify:\[ \frac{(a^2 - b^2)\text{sinh}(b) + 2abi\text{cosh}(b)}{2abi e^{-ia}} \]Separate the real and imaginary parts:\[ \frac{(a^2 - b^2)\text{sinh}(b)}{2abi e^{-ia}} + \frac{2abi\text{cosh}(b)}{2abi e^{-ia}} \]Simplify each term:\[ \frac{(a^2 - b^2)\text{sinh}(b)}{2abe^{-ia}} + \text{cosh}(b)e^{ia} \]

Evaluate the Absolute Square

Find the modulus of the complex expression:\[ \bigg|\frac{(a^2 - b^2)\text{sinh}(b)}{2abe^{-ia}} + \text{cosh}(b)e^{ia}\bigg|^{2} \]Evaluate each part separately using properties of modulus for complex numbers:

Use Properties of Complex Modulus

Use the properties \(|z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 + 2\text{Re}(z_1\bar{z_2})\) (if need be, simplify before squaring):

Final Step: Combine Results

Combine all terms and simplify to reach the final result.

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that explains the behavior of particles at the atomic and subatomic levels. It incorporates complex numbers extensively to describe various quantum states. In our problem, the given complex expression is typical in quantum mechanics as it can represent a wave function or probability amplitude. The square of the modulus of a complex number often gives probabilities or expectations in quantum systems.
Understanding how to manipulate and simplify complex expressions is crucial in quantum mechanics. The given exercise helps deepen this skill, which is pivotal for solving real-world physics problems.

Hyperbolic Functions

Hyperbolic functions such as sinh (hyperbolic sine) and cosh (hyperbolic cosine) are analogs of the trigonometric functions but are related to a hyperbola rather than a circle. These functions arise in various areas of mathematics, including the study of complex numbers.
In our exercise, hyperbolic functions help simplify the expression involving exponential functions. We utilize the identities:
\[ \text{sinh}(x) = \frac{e^x - e^{-x}}{2} \]
\[ \text{cosh}(x) = \frac{e^x + e^{-x}}{2} \]
By substituting these into the problem, we can reduce the expression into a more manageable form.
Hyperbolic functions help bridge the gap between exponential expressions and simpler trigonometric-like functions, making complex calculations more tractable.

Complex Modulus

The modulus of a complex number, also called the absolute value, measures its size or magnitude. For a complex number \( z = a + bi \), the modulus is given by
\[ |z| = \sqrt{a^2 + b^2} \]
In the exercise, we are asked to evaluate the absolute square of a complex expression. This involves determining the magnitude squared, which is useful for physical interpretations, such as probabilities, in quantum mechanics.
The properties of the modulus, such as \( |z_1 \times z_2| = |z_1| \times |z_2| \) and \( |z_1/z_2| = |z_1|/|z_2| \), allow us to simplify the given complex fractions effectively.

Complex Arithmetic

Complex arithmetic involves operations such as addition, subtraction, multiplication, and division of complex numbers. These operations follow specific rules:
\ * To add or subtract two complex numbers, combine their real parts and imaginary parts separately.
\ * For multiplication and division, use properties of i, where \(i^2 = -1\).
In our problem, we perform complex multiplication by expanding terms and simplifying the expression using identity properties. For instance, combining\( (a + bi)^2 \) and \( (a - bi)^2 \) requires careful handling of the real and imaginary components.
Understanding these rules is crucial for successfully manipulating complex expressions in various mathematical and physical contexts.

Imaginary Numbers

Imaginary numbers are numbers that can be written in the form \( bi \), where \( b \) is a real number and \( i \) is the imaginary unit, satisfying \( i^2 = -1 \). They extend the real numbers to form the complex numbers. In our exercise, terms involving \( bi \) appear throughout the expression.
Imaginary numbers are essential in various disciplines, particularly in electrical engineering, signal processing, and quantum mechanics. They allow for solutions to otherwise unsolvable equations and represent oscillatory phenomena.
The ability to manipulate and simplify expressions involving imaginary numbers is key to many advanced mathematical and physical theories, making problems like ours essential practice.