Question

Compute \(\mid \Psi \mid ^2 for \space \Psi = \psi \space sin \space \omega t\), where \(\psi\) is time independent and \(\omega\) is a real constant. Is this a wave function for a stationary state? Why or why not?

Short answer

\(|\Psi|^2 = \psi^2 \sin^2 \omega t\); not a stationary state as it's time-dependent.

Step-by-step solution

Find the complex conjugate of Ψ

The given function is \(\Psi = \psi \sin \omega t\). First, we identify if this function has any imaginary parts or real terms that would give a different complex conjugate. \(\psi\) is time-independent, and \(\sin \omega t\) is real, so the complex conjugate \(\Psi^*\) will be the same as \(\Psi\): \(\Psi^* = \psi \sin \omega t\).

Compute the magnitude squared, |Ψ|²

The magnitude squared of a complex function \(\Psi\) is given by the product of the function and its complex conjugate: \(|\Psi|^2 = \Psi \Psi^*\). Since \(\Psi = \psi \sin \omega t\), we have: \[|\Psi|^2 = (\psi \sin \omega t) (\psi \sin \omega t) = \psi^2 \sin^2 \omega t.\]

Analyze if this is a wave function for a stationary state

For a stationary state, \(|\Psi|^2\) should be independent of time. In this case, \(|\Psi|^2 = \psi^2 \sin^2 \omega t\), which is clearly dependent on time due to the \(\sin^2 \omega t\) factor. Thus, \(\Psi\) is not a wave function for a stationary state.

Key concepts

Wave Function

In quantum mechanics, a wave function, often symbolized as \(\Psi\), provides a mathematical description of the quantum state of a system. It encompasses all the information about a particle, including its position, momentum, and even its spin. The wave function is typically expressed as a complex-valued function, \(\Psi(x, t)\), which means it can have both real and imaginary components.

• In the context of the original exercise, \(\Psi\) is given as \(\psi \sin \omega t\).
• Here, \(\psi\) represents the time-independent part, while \(\sin \omega t\) is time-dependent and purely real.
• This highlights the dynamic nature of wave functions, crucial for understanding the evolution of quantum systems over time.

Stationary State

A stationary state is a special, stable state in quantum mechanics where the probability distribution doesn't change in time. This means that the magnitude squaared of the wave function, \(\mid \Psi \mid^2\), stays constant over time. In practical terms, this implies that the properties of the state do not evolve as time passes.

• In the problem at hand, we have \(\mid \Psi \mid^2 = \psi^2 \sin^2 \omega t\).
• Since \(\sin^2 \omega t\) is time-dependent, it causes \(\mid \Psi \mid^2\) to fluctuate over time.
• This fluctuation means that the state is not stationary, as time dependence is a crucial indicator of a non-stationary state.

Understanding this property helps in recognizing why certain quantum systems stay unchanging across time, while others might oscillate or evolve.

Complex Conjugate

A fundamental concept in dealing with complex numbers and functions, the complex conjugate, \(\Psi^*\), is obtained by changing the sign of the imaginary part of a complex number or function. In quantum mechanics, the complex conjugate is often used to calculate probabilities and to find the magnitude squared of a wave function.

• For the wave function \(\Psi = \psi \sin \omega t\), since \(\sin \omega t\) is a real function and \(\psi\) is time-independent, the complex conjugate is identical to the original: \(\Psi^* = \psi \sin \omega t\).
• This simplification is crucial for calculations, as it directly impacts how we determine the intensity or probability distribution of the quantum state.

Magnitude Squared

The magnitude squared of a wave function, expressed as \(\mid \Psi \mid^2\), reflects the probability density of finding a particle at a given position, at a specific time. In quantum mechanics, this concept is vital as it pertains to the physical interpretation of the wave function.

• For \(\Psi = \psi \sin \omega t\), the magnitude squared is calculated as \(\mid \Psi \mid^2 = \psi^2 \sin^2 \omega t\).
• This expression symbolizes that the probability density is a product of the spatial component, \(\psi^2\), and the time-varying term, \(\sin^2 \omega t\).
• Therefore, it's clear that the magnitude squared is not constant, indicating that the probability density fluctuates over time.

Understanding the concept of magnitude squared is pivotal in predicting the likelihood of a particle's presence in a particular region, which is a cornerstone of quantum theory.