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

Final result is: \ \left( \cosh(b) \right)^2 + \left( \frac{\sinh(b)(a^2 - b^2)}{2ab} \right)^2 \.

Step-by-step solution

- Simplify the numerator

In the numerator, express the complex terms explicitly: \((a + bi)^2 = a^2 + 2abi - b^2\) and \((a - bi)^2 = a^2 - 2abi - b^2\). Thus, the numerator becomes:\[ (a^2 + 2abi - b^2) e^b - (a^2 - 2abi - b^2) e^{-b} \]

- Combine like terms in the numerator

Combine the exponential terms and separate real and imaginary parts:\[ (a^2 + 2abi - b^2) e^b - (a^2 - 2abi - b^2) e^{-b} = a^2 (e^b - e^{-b}) + 2abi(e^b + e^{-b}) - b^2 (e^b - e^{-b}) \]

- Use hyperbolic identities

Recall the hyperbolic identities: \( e^b - e^{-b} = 2\sinh(b) \) and \( e^b + e^{-b} = 2\cosh(b) \). Substitute these into the expression:\[ a^2 (2\sinh(b)) + 2abi (2\cosh(b)) - b^2 (2\sinh(b)) \]Which can be simplified to:\[ 2a^2 \sinh(b) + 4abi \cosh(b) - 2b^2 \sinh(b) \]

- Simplify the numerator

Factor out the common factor of 2 in the numerator:\[ 2(a^2 \sinh(b) + 2abi \cosh(b) - b^2 \sinh(b)) = 2 \sinh(b) (a^2 - b^2) + 4abi \cosh(b) \]

- Construct the fraction

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

- Divide and simplify

Divide each term separately:\[ \frac{2 \sinh(b) (a^2 - b^2)}{4abi e^{-ia}} + \frac{4abi \cosh(b)}{4abi e^{-ia}} \]Simplify each term:\[ \frac{\sinh(b) (a^2 - b^2)}{2ab} e^{ia}i + \cosh(b) e^{ia} \]

- Simplify the expression further

Since \( e^{ia} \) is present in both terms, the expression inside the modulus becomes:\[ i \frac{\sinh(b) (a^2 - b^2)}{2ab} + \cosh(b) \e^{ia} \]Dropping the exponential term since for modulus it does not affect the result:\[ i \frac{\sinh(b) (a^2 - b^2)}{2ab} + \cosh(b) \]

- Evaluate the modulus squared

The modulus of a complex number \( z = x + yi \) is given by \( \sqrt{x^2 + y^2} \). Considering that our expression is \(z\) in the form \( A + Bi\), the modulus squared is:\[ \left( \cosh(b) \right)^2 + \left( \frac{\sinh(b)(a^2 - b^2)}{2ab} \right)^2 \].

Key concepts

absolute square

The concept of the absolute square involves finding the square of the absolute value of a complex number. This is often used in quantum mechanics to determine probabilities.

• The absolute value (or modulus) of a complex number \( z = x + yi \) is given by the formula \( |z| = \sqrt{x^2 + y^2} \).
• The absolute square is then obtained by squaring the modulus: \( |z|^2 = x^2 + y^2 \).
• It essentially measures the magnitude of the complex number ignoring the phase (angle) part.

In the context of the exercise, evaluating the absolute square helps in deriving meaningful physical quantities from complex expressions. It simplifies the expression by focusing on its magnitude.

complex number

Complex numbers extend the concept of one-dimensional real numbers to the two-dimensional complex plane by introducing a new unit called 'i', where \(i^2 = -1\).

• A complex number can be written as \( z = x + yi \), where \(x\) and \(y\) are real numbers.
• The part \(x\) is called the real part, and \(y\) is the imaginary part.
• Operations such as addition, subtraction, multiplication, and division can be performed on complex numbers, similar to real numbers, but following special rules involving 'i'.

In the given exercise, understanding the properties of complex numbers is essential to manipulate the numerator and denominator correctly.

hyperbolic function

Hyperbolic functions are analogs of trigonometric functions but for the hyperbola rather than the circle. They are important in various areas such as calculus, complex analysis, and physics.

• The basic hyperbolic functions are the hyperbolic sine \(\sinh(x)\) and hyperbolic cosine \(\cosh(x)\).
• Defined as \( \sinh(x) = \frac{e^x - e^{-x}}{2} \) and \( \cosh(x) = \frac{e^x + e^{-x}}{2} \), these functions share properties similar to sine and cosine functions.
• Hyperbolic identities, such as \( e^x - e^{-x} = 2\sinh(x) \) and \( e^x + e^{-x} = 2\cosh(x) \), are used for simplifying expressions involving exponentials.

In the exercise, we use these identities to simplify the expression involving \(e^b\) and \(e^{-b}\).

quantum mechanics

Quantum mechanics is the branch of physics that deals with the behavior of particles on an atomic and subatomic scale.

• It describes physical properties using probabilistic models and complex numbers.
• Key principles include wave-particle duality, quantization, and the uncertainty principle.
• Complex numbers and their properties, such as the absolute square, are used to calculate probabilities and physical observables.

In the given exercise, the expression you are working with arises in quantum mechanics to possibly relate to the probabilities or amplitudes of quantum states. The absolute square provides meaningful results, such as the probability density, which is crucial in quantum theory.