Question

Show that \(\Phi\)(\(\phi\)) = \(e$$^{im_l}$$^\phi\) = \(\Phi\)(\(\phi\) + 2\(\pi\)) (that is, show that \(\Phi\) (\(\phi\)) is periodic with period 2\(\pi\)) if and only if m\(_l\) is restricted to the values 0, \(\pm\)1, \(\pm\)2,.... (\(Hint\): Euler's formula states that \(e$$^i$$^\phi\) = cos \(\phi\) + \(i\) sin \(\phi\).)

Short answer

\(\Phi(\phi)\) is periodic with period \(2\pi\) if \(m_l\) is an integer (0, \(\pm1, \pm2,\ldots\)).

Step-by-step solution

Define Periodicity

A function is periodic with period \(2\pi\) if it satisfies the condition \(\Phi(\phi) = \Phi(\phi + 2\pi)\). Our goal is to verify this condition for the function \(\Phi(\phi) = e^{im_l\phi}\).

Apply the Periodicity Condition

Substitute \(\phi + 2\pi\) into \(\Phi\): \[\Phi(\phi + 2\pi) = e^{im_l(\phi + 2\pi)}\]This can be expanded using properties of exponents:\[= e^{im_l\phi} \cdot e^{im_l \cdot 2\pi}\]

Evaluate Euler's Formula for \(e^{im_l \cdot 2\pi}\)

Recognizing that \(e^{im_l \cdot 2\pi}\) can be expressed using Euler's formula, we have:\[e^{im_l \cdot 2\pi} = \cos(2\pi m_l) + i\sin(2\pi m_l)\]Since \(\cos(2\pi m_l) = 1\) and \(\sin(2\pi m_l) = 0\) for any integer \(m_l\), this simplifies to:\[e^{im_l \cdot 2\pi} = 1\]

Conclude the Periodicity

Substitute this back into the expression from Step 2:\[\Phi(\phi + 2\pi) = e^{im_l\phi} \cdot 1 = e^{im_l\phi} = \Phi(\phi)\]The equality \(\Phi(\phi) = \Phi(\phi + 2\pi)\) shows that \(\Phi(\phi)\) is indeed periodic with period \(2\pi\) for integer values of \(m_l\).

Restriction on \(m_l\) Values

The derivation relied on the condition \(e^{im_l \cdot 2\pi} = 1\), which holds true when \(m_l\) is an integer (since \(\cos(2\pi k) = 1, \sin(2\pi k) = 0\) for any integer \(k\)). Thus, \(m_l\) must be 0, \(\pm1, \pm2, \ldots\) to satisfy this condition.

Key concepts

Periodicity

Periodicity in mathematics and physics describes a repeating pattern over regular intervals. When we say a function is periodic, it means that the function's values repeat in a predictable manner as we progress through its domain, much like how the sequences of a melody might cycle in music. In the context of the provided exercise, we are focusing on the function \( \Phi(\phi) = e^{im_l\phi} \).
This function is periodic with a period of \(2\pi\) if it satisfies the equation \( \Phi(\phi) = \Phi(\phi + 2\pi) \). Simply put, every interval of \(2\pi\) gets us back to the starting point. When a function is \(2\pi\)-periodic, it essentially repeats its pattern in every successive interval of \(2\pi\).
For the function \( e^{im_l\phi} \), this periodicity holds true only when \(m_l\) is an integer. This is because the exponential function evaluates to 1 whenever the exponent is a whole multiple of \(2\pi i\). Hence, integers play a pivotal role in ensuring this periodicity across the function's domain.

Wave Functions

Wave functions are fundamental concepts in quantum mechanics which describe the probability amplitude of a particle's location and state. In essence, wave functions provide the probabilistic information about where or how a particle is observed at a given moment.
In the exercise at hand, the function \( \Phi(\phi) \) can be thought of as a type of wave function. Its periodic nature ties into the quantum mechanical behavior of particles, many of which exhibit wave-like characteristics. For instance, the "waves" may repeat at every \(2\pi\) interval.
This periodicity in wave functions is particularly important in analyzing systems with symmetrical properties, such as electrons in a hydrogen atom. These electrons exist within quantized orbitals which have wave functions exhibiting precise periodic patterns depending on their quantum states. These states are defined by quantum numbers, one of which is represented by \(m_l\) in the exercise. The values of \(m_l\), such as 0, \(\pm 1\), \(\pm 2\), etc., determine the symmetry and repeating characteristics of these wave functions.

Euler's Formula

Euler's formula is a remarkable equation bridging trigonometry and complex exponentials, expressed as \( e^{i\phi} = \cos(\phi) + i\sin(\phi) \). This formula is pivotal in simplifying and understanding the behavior of complex functions, especially those often found in wave mechanics and quantum physics.
In our exercise, Euler’s formula helps us understand why the expression \( e^{im_l \cdot 2\pi} \) equals 1 when \(m_l\) is an integer. Specifically, when an integer multiple of \(2\pi\) is placed in Euler's formula, \( \cos(2\pi k) = 1 \) and \( \sin(2\pi k) = 0 \) for any integer \(k\), confirming that \( e^{im_l \cdot 2\pi} = 1 \).
Consequently, Euler's formula does not just simplify calculations, it reveals fundamental properties of periodic functions. It highlights how rotations by full circles (\(2\pi\)) in the complex plane return to the starting point. Understanding this concept is crucial for students delving into quantum mechanics, as it illuminates how complex waveforms evolve and repeat, underpinning the periodic behaviors of wave functions.