Question

(a) Show that Re \(z=\frac{1}{2}(z+\bar{z})\) and that \(\operatorname{Im} z=(1 / 2 i)(z-\bar{z})\). (b) Show that \(\left|e^{z}\right|^{2}=e^{2 \mathrm{Re} z}\). (c) Use (b) to evaluate \(\left|e^{(1+i x)^{2}(1-i t)-|1+i t|^{2}}\right|^{2}\) which occurs in quantum mechanics.

Short answer

Re z = 1/2 (z + \bar{z}) and Im z = 1/2i (z - \bar{z}). |e^z|^2 = e^{2 Re z}. |e^{(1+ix)^2(1-it)-|1+it|^2}|^2 = e^{-2x^2 - 2t^2}.

Step-by-step solution

Express Real and Imaginary Parts of Complex Number

Consider a complex number z which can be written as: \[ z = a + bi \] where a and b are real numbers, and i is the imaginary unit.

Conjugate of a Complex Number

Write the conjugate of z: \[ \bar{z} = a - bi \]

Prove the Real Part Formula

Add z and its conjugate: \[ z + \bar{z} = (a + bi) + (a - bi) = 2a \] Dividing both sides by 2 gives: \[ \operatorname{Re}z = \frac{1}{2}(z + \bar{z}) \]

Prove the Imaginary Part Formula

Subtract z's conjugate from z: \[ z - \bar{z} = (a + bi) - (a - bi) = 2bi \] Divide both sides by 2i to get: \[ \operatorname{Im}z = \frac{1}{2i}(z - \bar{z}) \]

Rewriting the Exponential Function

For a complex number z, write its exponential: \[ e^z = e^{a + bi} = e^a e^{bi} \] Using Euler's formula, we have: \[ e^{bi} = \cos(b) + i\sin(b) \] Thus: \[ |e^z| = |e^a e^{bi}| = |e^a| |e^{bi}| = e^a |\cos(b) + i \sin(b)| = e^a \] since \( |\cos(b) + i \sin(b)| = 1 \)

Prove the Exponential Magnitude Formula

Thus: \[ |e^z|^2 = (e^a)^2 = e^{2a} = e^{2 \operatorname{Re}z} \] confirming the required formula.

Applying the Formula

Let z be: \[ z = (1 + ix)^2(1 - it) - |1 + it|^2 \] First calculate each part: \[ |1 + it|^2 = (1 + it)(1 - it) = 1 + t^2 \] Now, expand the squared term: \[ (1 + ix)^2 = 1 + 2ix - x^2 \] Therefore, \[ z = (1 + 2ix - x^2)(1 - it) - (1 + t^2) \] Expand and simplify: \[ z = 1 + 2ix - x^2 - it + 2itx - x^2t - (1 + t^2) \] Combining similar terms: \[ z = (1 - 1) - x^2 - t^2 + i(2x - t + 2tx - x^2t) \] The real part is: \[ \operatorname{Re} z = -x^2 - t^2 \]

Final Calculation of the Magnitude

Using the previously proved formula: \[ \left|e^{(1+ix)^2 - |1 + it|^2}\right|^2 = e^{2 \operatorname{Re}((1+ix)^2(1-it) - |1+it|^2)} = e^{2 (-x^2 - t^2)} = e^{-2x^2 - 2t^2} \]

Key concepts

Real Part of Complex Numbers

A complex number is typically written in the form \(z = a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part. To extract the real part of a complex number, we can use the formula:

\[\text{Re}(z) = \frac{1}{2}(z + \bar{z})\]

Here, the complex conjugate \(\bar{z}\) is defined as \(\bar{z} = a - bi\). By adding \(z\) and its conjugate, the imaginary parts cancel out, leaving only the real part:

\[z + \bar{z} = (a + bi) + (a - bi) = 2a\]

Dividing by 2 gives us the real part:

\[\text{Re}(z) = \frac{1}{2}(z + \bar{z}) = a\]

This formula is useful for separating the real part of a complex number in various mathematical operations.

Imaginary Part of Complex Numbers

The imaginary part of a complex number \(z = a + bi\) can also be separated using a specific formula:

\[\text{Im}(z) = \frac{1}{2i}(z - \bar{z})\]

To derive this, we start by subtracting the conjugate of \(z\) from \(z\):

\[z - \bar{z} = (a + bi) - (a - bi) = 2bi\]

Dividing both sides by \(2i\) results in:

\[\text{Im}(z) = \frac{1}{2i}(z - \bar{z}) = b\]

This formula is crucial whenever we need to isolate the imaginary part in complex number operations.

Exponential Functions of Complex Numbers

Exponential functions involving complex numbers are fundamentally different from real exponentials. For a complex number \(z = a + bi\), the exponential function is:

\[e^z = e^{a + bi} = e^a \times e^{bi}\]

Using Euler's formula, \(e^{bi} = \text{cos}(b) + i\text{sin}(b)\), we can simplify the expression:

\[e^z = e^a (\text{cos}(b) + i\text{sin}(b))\]

To find the magnitude of this expression, note that:

\[|e^z| = |e^a (\text{cos}(b) + i\text{sin}(b))| = |e^a| |\text{cos}(b) + i\text{sin}(b)|\]

Since the magnitude of \(\text{cos}(b) + i\text{sin}(b)\) is always 1:

\[|e^z| = e^a\]

Thus, when squared, we obtain:

\[|e^z|^2 = (e^a)^2 = e^{2a} = e^{2 \text{Re}(z)}\]

This result is foundational in fields like complex analysis and quantum mechanics.

Quantum Mechanics

Quantum mechanics often deals with complex numbers. They are pivotal in describing the state functions and probabilities.

For instance, the formula for exponential functions of complex numbers plays a role in quantum mechanics problems. Consider the expression

\[|e^{(1 + ix)^2(1 - it) - |1 + it|^2}|^2\]

Using the principles from exponential functions of complex numbers, we can simplify this problem. First, compute the real part of the expression, then square it and apply:

\[|e^{z}|^2 = e^{2 \text{Re}(z)}\]

Working through the steps in the problem, we eventually find that:

\[ e^{-2x^2 - 2t^2}\]

This simplification eases the computation in quantum mechanical contexts, which often involve complex numbers in wave functions and probability amplitudes.

Complex Conjugate

The complex conjugate of a complex number \(z = a + bi\) is denoted as \(\bar{z}\) and is defined as:

\[\bar{z} = a - bi\]

Taking the complex conjugate essentially flips the sign of the imaginary part. This operation is extremely useful in various mathematical manipulations:

• Isolating real and imaginary parts.
• Simplifying expressions involving complex numbers.
• Calculating magnitudes and performing divisions.

When dealing with conjugates, remember that:

\[z \times \bar{z} = (a + bi)(a - bi) = a^2 + b^2\]

This product always results in a non-negative real number. The complex conjugate is a staple concept in fields like signal processing, quantum mechanics, and electrical engineering.