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Chapter 7: Problem 45

What is the physical significance of the wave function?

Short Answer

Expert verified
The wave function in quantum mechanics, denoted by the Greek letter 'psi' (\(\Psi\)), is a mathematical description that encapsulates the quantum state of a system. Its physical significance lies in the Born Rule, where the square of the magnitude of the wave function (\(|\Psi|^2\)) gives the probability density of finding the particle at a certain location. The wave function also represents the principle of superposition, suggesting that a particle simultaneously exists in all theoretically possible states.

Step by step solution

01

Introduction to Wave Function

In quantum mechanics, a wave function, \(\Psi\), is a mathematical function that describes the quantum state of a particle. It is a fundamental concept in the field of quantum mechanics.
02

Probability Density

The physical interpretation of the wave function is given by the Born Rule, where the square of the magnitude of the wave function, \(|\Psi|^2\), at a certain point gives the probability density of finding the particle at that location. Thus, if the wave function itself is a complex number, its modulus squared gives a real number that can be interpreted as a probability density.
03

Superposition of States

An important feature of quantum mechanics is the principle of superposition, which implies that a physical system - such as an electron in an atom - simultaneously exists in all its theoretically possible states. The wave function provides a way to calculate the probability of observing each of these states. The state of any quantum system is a superposition of all possible states.

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Most popular questions from this chapter

Indicate which of the following sets of quantum numbers in an atom are unacceptable and explain why: (a) \(\left(1,0, \frac{1}{2}, \frac{1}{2}\right)\) (b) \(\left(3,0,0,+\frac{1}{2}\right)\) (c) \(\left(2,2,1,+\frac{1}{2}\right)\) (d) \(\left(4,3,-2,+\frac{1}{2}\right)\), (e) \((3,2,1,1)\).

What is the maximum number of electrons in an atom that can have the following quantum numbers? Specify the orbitals in which the electrons would be found. (a) \(n=2, m_{s}=+\frac{1}{2} ;\) (b) \(n=4, m_{\ell}=+1\) (c) \(n=3, \ell=2 ;\) (d) \(n=2, \ell=0, m_{s}=-\frac{1}{2} ;\) (e) \(n=4\) ,\(\ell=3, m_{\ell}=-2\)

The UV light that is responsible for tanning the skin falls in the 320 - to 400 -nm region. Calculate the total energy (in joules) absorbed by a person exposed to this radiation for \(2.0 \mathrm{~h}\), given that there are \(2.0 \times\) \(10^{16}\) photons hitting Earth's surface per square centimeter per second over a \(80-\mathrm{nm}(320 \mathrm{nm}\) to \(400 \mathrm{nm})\) range and that the exposed body area is \(0.45 \mathrm{~m}^{2}\). Assume that only half of the radiation is absorbed and the other half is reflected by the body. (Hint: Use an average wavelength of \(360 \mathrm{nm}\) in calculating the energy of a photon.)

The sun is surrounded by a white circle of gaseous material called the corona, which becomes visible during a total eclipse of the sun. The temperature of the corona is in the millions of degrees Celsius, which is high enough to break up molecules and remove some or all of the electrons from atoms. One way astronomers have been able to estimate the temperature of the corona is by studying the emission lines of ions of certain elements. For example, the emission spectrum of \(\mathrm{Fe}^{14+}\) ions has been recorded and analyzed. Knowing that it takes \(3.5 \times 10^{4} \mathrm{~kJ} / \mathrm{mol}\) to convert \(\mathrm{Fe}^{13+}\) to \(\mathrm{Fe}^{14+},\) estimate the temperature of the sun's corona.

Which orbital in each of the following pairs is lower in energy in a many- electron atom? (a) \(2 s, 2 p\) (b) \(3 p, 3 d ;\) (c) \(3 s, 4 s ;\) (d) \(4 d, 5 f\).
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