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

Using Cramer's rule, \ A=\frac{K + i k}{K - i k}\. Verification shows \ |A|^2 = 1\.

Step-by-step solution

- Write the system of equations in matrix form

Represent the given equations in the form \ \AX = B\, where A is the matrix of coefficients, X is the vector of the unknowns, and B is the vector of constants.The given system of equations is:\[ \begin{aligned} A - B &= -1 \ i k A - K B &= i k \end{aligned} \]This can be written as:\[ \begin{aligned} \begin{pmatrix} 1 & -1 \ i k & -K \end{pmatrix} \begin{pmatrix} A \ B \end{pmatrix} = \begin{pmatrix} -1 \ i k \end{pmatrix} \end{aligned} \]

- Apply Cramer's rule

Cramer's rule involves calculating determinants to find the solutions for the unknowns. Let's denote the coefficient matrix as M and the vector of constants as C.\[ M = \begin{pmatrix} 1 & -1 \ i k & - K \end{pmatrix}, \quad X = \begin{pmatrix} A \ B \end{pmatrix}, \quad C = \begin{pmatrix} -1 \ i k \end{pmatrix} \]We need to calculate the determinant of M (denoted as det(M)) and the determinants of matrices where one column of M is replaced by C (denoted as det(M_A) for finding A).\( \text{det}(M) = 1 \times (-K) - (-1) \times (i k) = -K + i k \)

- Calculate det(M_A)

To find det(M_A), replace the first column of M with vector C:\[ M_A = \begin{pmatrix} -1 & -1 \ i k & - K \end{pmatrix} \]\( \text{det}(M_A) = -1 \cdot (-K) - (-1) \cdot (i k) = K + i k \)

- Solve for A using Cramer's rule

According to Cramer's rule, the solution for A is given by:\[ A = \frac{\text{det}(M_A)}{\text{det}(M)} = \frac{K + i k}{i k - K} \]We can simplify this expression by multiplying numerator and denominator by \ -1 \, giving us:\[ A = \frac{-(K + i k)}{-(i k - K)} = \frac{K + i k}{K - i k} \]This can be further simplified by multiplying the numerator and denominator by the conjugate of the denominator (K + i kd K - i kd):\[ A = \frac{K + i k}{K - i k} \cdot \frac{K + i k}{K + i k} = \frac{(K + i k)^2}{K^2 + (i k)^2} = \frac{K^2 + 2 i k K - k^2}{K^2 - k^2} \]Since i k^2 = -k^2 ,\[ A = \frac{K^2 + 2 i k K - k^2}{K^2 - k^2} \]

- Verify that |A|^2=1

To verify that \ |A|^2=1, let's calculate \ |A|^2 \, which is \ A \times A^* \, where \ A^* \ is the complex conjugate of A.\[ A = \frac{K + i k}{K - i k} \]The complex conjugate of this is:\[ A^* = \frac{K - i k}{K + i k} \]Now, \[ |A|^2 = A \cdot A^* = \frac{K + i k}{K - i k} \cdot \frac{K - i k}{K + i k} = \frac{(K + i k) (K - i k)}{(K - i k)(K + i k)} = \frac{K^2 + k^2}{K^2 + k^2} = 1 \]

Key concepts

Understanding Matrix Equations

A matrix equation is an efficient way to represent and solve multiple linear equations simultaneously. Instead of writing out each equation, you represent the system using matrices, which simplifies the problem.
Let's break it down:

• Each row of the coefficient matrix represents the coefficients of one equation.
• The column matrix of variables represents the unknowns you are solving for.
• The constants are placed in a separate column matrix, representing the result of each equation.

In the given example, the matrix equation is shown as:
\[ \begin{pmatrix} 1 & -1 \ i k & -K \end{pmatrix} \begin{pmatrix} A \ B \end{pmatrix} = \begin{pmatrix} -1 \ i k \end{pmatrix} \]
Here, the first matrix is the coefficient matrix, the second matrix contains the variables, and the third matrix represents the constants.
This compact representation allows us to apply methods like Cramer's Rule more easily.
Understanding this concept is fundamental to solving more complex systems in mathematics and physics.

Complex Numbers in Quantum Mechanics

Complex numbers are essential in quantum mechanics for representing wave functions and probabilities. A complex number is a number of the form \(a + bi\), where \(i = \sqrt{-1}\).
In our problem, we use \(i\), the imaginary unit, to represent complex interactions.
There are a few important properties of complex numbers relevant here:

• Addition and subtraction: You can add or subtract the real and imaginary parts separately.
• Multiplication: Use distributive property, keeping in mind \(i^2 = -1\).
• Conjugation: The conjugate of \(a + bi\) is \a - bi\. Multiplying a complex number by its conjugate results in a real number.

In the provided solution, we encounter terms like \(i k\), emphasizing how complex numbers handle oscillations and rotations—a key feature in quantum mechanics.
Mastering complex numbers is crucial for understanding the mathematical frameworks used in quantum theory and other advanced fields.

Determinants and Cramer's Rule

Determinants are numbers calculated from a square matrix of any order. They have significant properties and applications in solving linear systems of equations and finding matrix inverses.
To calculate the determinant for a 2x2 matrix:
\[ \begin{pmatrix} a & b \ c & d \end{pmatrix} \text{det} = ad - bc \]
In Cramer's Rule, we use determinants to solve for variables in a system of linear equations.

• Calculate the determinant of the coefficient matrix (det(M)).
• Replace one column of the coefficient matrix with the constants vector to form a new matrix.
• Calculate the determinant of this new matrix for each variable.
• Divide the determinant of the new matrix by the determinant of the original coefficient matrix to find the value of the variable.

In our example, we compute:\[ \text{det}(M) = 1 \times (-K) - (-1) \times (i k) = -K + i k \]
By following these steps, we solve for \(A\) and verify \|A\|^2 = 1, demonstrating the effectiveness of determinants in finding solutions.
A deep understanding of determinants will strengthen your ability to tackle a wide range of mathematical problems.