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Chapter 7: Problem 13

Suppose a wave function is discontinuous at some point. Can this function represent a quantum state of some physical particle? Why? Why not?

Short Answer

Expert verified
A discontinuous wave function cannot represent a quantum state of a physical particle because it does not satisfy the requirements of normalization and continuity. Normalization ensures the total probability of finding the particle is equal to 1, and continuity allows us to calculate the momentum and kinetic energy from the function's derivatives. Since a discontinuous wave function does not meet these conditions, it cannot describe a quantum state of any physical particle.

Step by step solution

01

Wave Function in Quantum Mechanics

In quantum mechanics, a wave function describes the quantum state of particles in a system. The wave function is a mathematical function that contains information about the probability of finding a particle in a particular state. It plays an important role in understanding the behavior of particles at the quantum level.
02

Normalization of Wave Functions

Normalization is a key property of a wave function. It ensures that the total probability of finding the particle in its entire space is equal to 1, which means the particle must be somewhere in the system. To normalize a wave function, we take the absolute square of the function (also known as the probability density) and integrate it over the entire space. This normalization condition can be mathematically expressed as: \[\int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1\] Here, \(\psi(x)\) is the wave function and \(|\psi(x)|^2\) is the probability density of finding the particle in a specific region.
03

Continuity of Wave Functions

In general, wave functions must be continuous, meaning there are no sudden jumps or gaps in the function as we move through its domain. The reason for this requirement is related to the normalization condition. If the wave function is discontinuous at some point, the probability density (the absolute square of the wave function) will not be well-defined at that point, and we will not be able to integrate it over the entire space to normalize it. Moreover, if a wave function is discontinuous, we cannot compute its first and second derivatives, which are needed to calculate the momentum and kinetic energy of the particle, respectively.
04

Discontinuous Wave Function as a Representation of Quantum State

Given the requirements of normalization and continuity, we can now say that a discontinuous wave function cannot represent a quantum state of a physical particle. The discontinuity makes it impossible to normalize the wave function, as well as to calculate its momentum and kinetic energy from the first and second derivatives of the function. Therefore, a wave function must be continuous and normalized to properly describe a quantum state of a physical particle. A discontinuous wave function cannot satisfy these requirements, and hence cannot represent a quantum state of any physical particle.

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Most popular questions from this chapter

Explain the connection between Planck's hypothesis of energy quanta and the energies of the quantum harmonic oscillator.

Consider an infinite square well with wall boundaries \(x=0 \quad\) and \(\quad x=L . \quad\) Explain \(\quad\) why the function \(\psi(x)=A \cos k x \quad\) is not a solution to the stationary Schrödinger equation for the particle in a box.

Compute \(\quad|\Psi(x, t)|^{2} \quad\) for \(\quad\) the function \(\Psi(x, t)=\psi(x) \sin \omega t,\) where \(\omega\) is a real constant.

94\. A particle of mass \(m\) confined to a box of width \(L\) is in its first excited state \(\psi_{2}(x) .\) (a) Find its average position (which is the expectation value of the position). (b) Where is the particle most likely to be found?

For a quantum particle in a box, the first excited state \(\left(\Psi_{2}\right)\) has zero value at the midpoint position in the box, so that the probability density of finding a particle at this point is exactly zero. Explain what is wrong with the following reasoning: "If the probability of finding a quantum particle at the midpoint is zero, the particle is never at this point, right? How does it come then that the particle can cross this point on its way from the left side to the right side of the box?
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