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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

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

Step by step solution

01

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 \(\Psi\): \(\Psi^* = \psi \sin \omega t\).
02

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: \[|\Psi|^2 = (\psi \sin \omega t) (\psi \sin \omega t) = \psi^2 \sin^2 \omega t.\]
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Consider a particle moving in one dimension, which we shall call the \(x\)-axis. (a) What does it mean for the wave function of this particle to be \(normalized\)? (b) Is the wave function \(\psi(x) = e^{ax}\) , where a is a positive real number, normalized? Could this be a valid wave function? (c) If the particle described by the wave function \(\psi(x) = Ae^{-bx}\), where \(A\) and \(b\) are positive real numbers, is confined to the range \(x \geq 0\), determine A (including its units) so that the wave function is normalized.

A particle of mass \(m\) in a one-dimensional box has the following wave function in the region \(x\) = 0 to \(x = L\) : $$\Psi(x, t) = {1\over \sqrt2} \psi_1(x)e^{-iE_1t/\hslash} + {1\over \sqrt 2} \psi_3(x)e^{-iE_3t/\hslash}$$ Here \(\psi_1(x)\) and \(\psi_3(x)\) are the normalized stationary-state wave functions for the \(n\) = 1 and \(n\) = 3 levels, and \(E_1\) and \(E_3\) are the energies of these levels. The wave function is zero for \(x <\) 0 and for \(x > L\). (a) Find the value of the probability distribution function at \(x = L\)/2 as a function of time. (b) Find the angular frequency at which the probability distribution function oscillates.

Consider the wave packet defined by $$\psi(x) = \int ^\infty_0 B(k)cos kx dk$$ Let \(B(k) = e^{-a^2k^2}\). (a) The function \(B(k)\) has its maximum value at \(k\) = 0. Let \(k_h\) be the value of \(k\) at which \(B(k)\) has fallen to half its maximum value, and define the width of \(B(k)\) as \(w_k = k_h\) . In terms of \(\alpha\), what is \(w_k\) ? (b) Use integral tables to evaluate the integral that gives \(\psi(x)\). For what value of \(x\) is \(\psi(x)\) maximum? (c) Define the width of \(\psi(x)\) as \(w_x = x_h\) , where \(x_h\) is the positive value of \(x\) at which \(\psi(x)\) has fallen to half its maximum value. Calculate \(w_x\) in terms of \(\alpha\). (d) The momentum \(p\) is equal to \(hk/2\pi\), so the width of \(B\) in momentum is \(w_p = hw_k /2\pi\). Calculate the product \(w_p w_x\) and compare to the Heisenberg uncertainty principle.

The ground-state energy of a harmonic oscillator is 5.60 eV. If the oscillator undergoes a transition from its \(n\) = 3 to \(n\) = 2 level by emitting a photon, what is the wavelength of the photon?

A harmonic oscillator consists of a 0.020-kg mass on a spring. The oscillation frequency is 1.50 Hz, and the mass has a speed of 0.480 m/s as it passes the equilibrium position. (a) What is the value of the quantum number n for its energy level? (b) What is the difference in energy between the levels \(E_n\) and \(E_{n+1}\)? Is this difference detectable?

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