Question

In the following set of equations (from a quantum mechanics problem), \(A\) and \(B\) are the unknowns, \(k\) and \(K\) are given, and \(i=\sqrt{-1}\). Use Cramer's rule to find \(A\) and show that \(|A|^{2}=1\) $$\left\\{\begin{aligned} A-B &=-1 \\ i k A-K B &=i k \end{aligned}\right.$$

Short answer

|A|^2 = 1

Step-by-step solution

Write the system in matrix form

Rewrite the given system of equations in matrix form. This can be written as: \[ \begin{bmatrix} 1 & -1 \ i k & -K \end{bmatrix} \begin{bmatrix} A \ B \end{bmatrix} = \begin{bmatrix} -1 \ i k \end{bmatrix} \]

Find the determinant of the coefficient matrix

Find the determinant of the coefficient matrix: \[ \text{det}(M) = \begin{vmatrix} 1 & -1 \ i k & -K \end{vmatrix} = (1)(-K) - (-1)(ik) = -K + ik \]

Replace the first column with the constants

Create a new matrix with the first column replaced by the constants vector. \[ M_A = \begin{bmatrix} -1 & -1 \ i k & -K \end{bmatrix} \]

Find the determinant of the new matrix

Find the determinant of the new matrix \[ \text{det}(M_A) = \begin{vmatrix} -1 & -1 \ i k & -K \end{vmatrix} = (-1)(-K) - (-1)(ik) = K - ik \]

Apply Cramer's rule to solve for A

Using Cramer's rule, solve for \(A\): \[ A = \frac{\text{det}(M_A)}{\text{det}(M)} = \frac{K - ik}{-K + ik} \]

Simplify the expression for A

Simplify the fraction to express \(A\) in simpler terms. Multiply the numerator and denominator by the conjugate of the denominator: \[ A = \frac{(K - ik)(-K - ik)}{(-K + ik)(-K - ik)} = \frac{-K^2 - iKk + iKk - i^2k^2}{K^2 + i^2k^2} = \frac{-K^2 + k^2}{K^2 + k^2}\] Since \(i^2 = -1\)

Verify \(|A|^2 = 1\)

\[ |A|^2 = \left| \frac{-K^2 + k^2}{K^2 + k^2} \right|^2 = \frac{(-K^2 + k^2)^2}{(K^2 + k^2)^2} = 1 \]

Key concepts

Quantum Mechanics

Quantum Mechanics is a branch of physics that deals with the behavior and interactions of particles at the quantum level. It is fundamentally different from classical mechanics in that it describes physical quantities in terms of probabilities rather than deterministic values. In Quantum Mechanics, particles such as electrons and photons exhibit both wave and particle properties. This duality is a core concept of Quantum Mechanics and is explained by the wave function, which encodes the probabilities of finding a particle in a particular state.

• Probabilities: Outcomes are not certain but given in terms of likelihoods.
• Wave-Particle Duality: Particles can behave like waves.
• Wavefunction: Describes the quantum state of a particle.

Understanding these principles helps in solving complex quantum systems like the given set of equations using techniques such as Cramer's Rule.

Matrix Algebra

Matrix Algebra is the branch of mathematics that deals with matrices and operations on them. Matrices are rectangular arrays of numbers or symbols arranged in rows and columns. Matrix Algebra is a fundamental tool in various fields including Quantum Mechanics, where it helps in representing and solving systems of linear equations.
In our exercise, we represent the system of equations as a matrix, forming a coefficient matrix and a constants vector. This aids in simplifying complex equations into a manageable format.
Key concepts in Matrix Algebra include matrix multiplication, determinants, and the inverse of a matrix. These tools allow for effective manipulation and solution of linear systems.

Complex Numbers

Complex Numbers are numbers that have a real part and an imaginary part, typically written in the form a + bi, where a and b are real numbers and i is the imaginary unit (\(i = \sqrt{-1}\). They are crucial in Quantum Mechanics, as many physical quantities are described using complex numbers.
For instance, in our system of equations, the coefficient matrix and constants vector contain complex numbers. This requires us to handle complex arithmetic operations such as addition, subtraction, multiplication, and finding the complex conjugate. Understanding how to work with complex numbers allows us to solve equations involving complex coefficients efficiently.

Determinants in Linear Algebra

The Determinant is a scalar value that is computed from the elements of a square matrix and provides important properties of the matrix. In Linear Algebra, the determinant is used to determine whether a matrix is invertible and to solve systems of linear equations.
In our problem, we calculate the determinants of the original coefficient matrix and the modified matrix. The determinant of the coefficient matrix is crucial for applying Cramer's Rule, which requires the determinant to be non-zero to ensure a unique solution exists for the system of equations.
Determinants also help in understanding geometric transformations described by matrices, such as scaling and rotation.

Cramer's Rule

Cramer's Rule is a powerful method in Linear Algebra for solving systems of linear equations with as many equations as unknowns. It expresses each unknown variable as a ratio of determinants. Specifically, each variable is obtained by dividing the determinant of a modified coefficient matrix (in which one column is replaced by the constants vector) by the determinant of the original coefficient matrix.
Applying Cramer's Rule involves:

• Writing the system of equations in matrix form.
• Computing the determinant of the coefficient matrix.
• Replacing columns of the matrix and computing new determinants.
• Solving for each variable by forming the ratio of determinants.

In our exercise, we use Cramer's Rule to solve for the variable A and verify that the modulus squared of A equals 1, a requirement in quantum mechanical contexts.