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By the end of this section, you will be able to:
  • Describe the statistical interpretation of the wave function
  • Use the wave function to determine probabilities
  • Calculate expectation values of position, momentum, and kinetic energy

In the preceding chapter, we saw that particles act in some cases like particles and in other cases like waves. But what does it mean for a particle to “act like a wave”? What precisely is “waving”? What rules govern how this wave changes and propagates? How is the wave function used to make predictions? For example, if the amplitude of an electron wave is given by a function of position and time, Ψ ( x , t ) , defined for all x , where exactly is the electron? The purpose of this chapter is to answer these questions.

Using the wave function

A clue to the physical meaning of the wave function Ψ ( x , t ) is provided by the two-slit interference of monochromatic light ( [link] ). (See also Electromagnetic Waves and Interference .) The wave function    of a light wave is given by E ( x , t ), and its energy density is given by | E | 2 , where E is the electric field strength. The energy of an individual photon depends only on the frequency of light, ε photon = h f , so | E | 2 is proportional to the number of photons. When light waves from S 1 interfere with light waves from S 2 at the viewing screen (a distance D away), an interference pattern is produced (part (a) of the figure). Bright fringes correspond to points of constructive interference of the light waves, and dark fringes correspond to points of destructive interference of the light waves (part (b)).

Suppose the screen is initially unexposed to light. If the screen is exposed to very weak light, the interference pattern appears gradually ( [link] (c), left to right). Individual photon hits on the screen appear as dots. The dot density is expected to be large at locations where the interference pattern will be, ultimately, the most intense. In other words, the probability (per unit area) that a single photon will strike a particular spot on the screen is proportional to the square of the total electric field, | E | 2 at that point. Under the right conditions, the same interference pattern develops for matter particles, such as electrons.

Part a shows monochromatic light of wavelength lambda emitted from a source, arriving as plane waves at a single slit, S. The waves pass through the slit ad form circular waves that arrive at a double slit, S sub 1 and S sub 2. The light rays emerge from two slits as semicircles overlapping one another. The interacting waves spread out and end on a screen where points of maximum, where the crests or troughs overlap, and minimum, where the crests from one slit overlap the troughs from the other, are marked. The pattern appears on the screen as a series of alternating bright and dark fringes. The fringes separation, y, is the distance between adjacent maxima. In part b, a photograph of the fringe pattern is shown. Part c shows how the pattern develops in time. Photos of the image at five times are shown. At first, only a few scattered bright points appear, apparently randomly, against a dark background. In the second image, we see more dots but not yet any discernible pattern. In the third image, we start to see that there are more dots in some parts of the image and fewer elsewhere. Vertical stripes of dense bright dots separated are clearly seen in the fourth image, and even more clearly in the fifth.
Two-slit interference of monochromatic light. (a) Schematic of two-slit interference; (b) light interference pattern; (c) interference pattern built up gradually under low-intensity light (left to right).

Visit this interactive simulation to learn more about quantum wave interference.

The square of the matter wave | Ψ | 2 in one dimension has a similar interpretation as the square of the electric field | E | 2 . It gives the probability that a particle will be found at a particular position and time per unit length, also called the probability density    . The probability ( P ) a particle is found in a narrow interval ( x , x + dx ) at time t is therefore

P ( x , x + d x ) = | Ψ ( x , t ) | 2 d x .

(Later, we define the magnitude squared for the general case of a function with “imaginary parts.”) This probabilistic interpretation of the wave function is called the Born interpretation    . Examples of wave functions and their squares for a particular time t are given in [link] .

Questions & Answers

For the question about the scuba instructor's head above the pool, how did you arrive at this answer? What is the process?
Evan Reply
as a free falling object increases speed what is happening to the acceleration
Success Reply
of course g is constant
acceleration also inc
which paper will be subjective and which one objective
photo electrons doesn't emmit when electrons are free to move on surface of metal why?
Rafi Reply
What would be the minimum work function of a metal have to be for visible light(400-700)nm to ejected photoelectrons?
Mohammed Reply
give any fix value to wave length
40 cm into change mm
Arhaan Reply
40cm=40.0×10^-2m =400.0×10^-3m =400mm. that cap(^) I have used above is to the power.
i.e. 10to the power -2 in the first line and 10 to the power -3 in the the second line.
there is mistake in my first msg correction is 40cm=40.0×10^-2m =400.0×10^-3m =400mm. sorry for the mistake friends.
40cm=40.0×10^-2m =400.0×10^-3m =400mm.
this msg is out of mistake. sorry friends​.
what is physics?
sisay Reply
why we have physics
Anil Reply
because is the study of mater and natural world
because physics is nature. it explains the laws of nature. some laws already discovered. some laws yet to be discovered.
is this a physics forum
Physics Reply
explain l-s coupling
Depk Reply
how can we say dirac equation is also called a relativistic equation in one word
preeti Reply
what is the electronic configration of Al
usman Reply
what's the signeficance of dirac equetion.?
Sibghat Reply
what is the effect of heat on refractive index
Nepal Reply
As refractive index depend on other factors also but if we supply heat on any system or media its refractive index decrease. i.e. it is inversely proportional to the heat.
you are correct
law of multiple
if we heated the ice then the refractive index be change from natural water
can someone explain normalization condition
Priyojit Reply
please tell
1 millimeter is How many metres
Darling Reply
1millimeter =0.001metre

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