The first step is to write down the wavefunction. You can measure a particle’s position and get a distinct value, but if you performed the measurement again in the exact same circumstances, you would get a different result. \nonumber \end{align*} \], Therefore, the expectation value of momentum is, \begin{align*} \langle p \rangle &= \int_0^L dx \left(Ae^{+i\omega t}sin \dfrac{\pi x}{L}\right)\left(-i \dfrac{Ah}{2L} e^{-i\omega t} cos \, \dfrac{\pi x}{L}\right) \nonumber \\[4pt] &= -i \dfrac{A^2h}{4L} \int_0^L dx \, \sin \, \dfrac{2\pi x}{L} \nonumber \\[4pt] &= 0. Individual photon hits on the screen appear as dots. (An odd function is also referred to as an anti-symmetric function.) The probability really relates to whether you're in world A, B or C, not where the particle is within your world. Quanta Magazine: New Support for Alternative Quantum View, University of Oregon: Copenhagen Interpretation, University of Virginia: General Uncertainty Principle, Georgia State University Hyper Physics: Schrodinger Equation, Georgia State University Hyper Physics: Expectation Values, LibreTexts: Deriving the de Broglie Wavelength. Soon, you will learn soon that the wavefunction can be used to make many other kinds of predictions, as well. If the screen is exposed to very weak light, the interference pattern appears gradually (Figure $$\PageIndex{1c}$$, left to right). The radioactive decay is a random [probabilistic] process, and there is no way to predict when it will happen. The momentum operator in the x-direction is sometimes denoted, \[\langle p \rangle = - i\hbar \dfrac{d}{dx},\label{7.10}, Momentum operators for the y- and z-directions are defined similarly. However, there is good news for physics students hoping to be able to pass classes in quantum mechanics. What rules govern how this wave changes and propagates? The procedure for doing this is, $\langle p \rangle = \int_{-\infty}^{\infty} \Psi^* (x,t) \, \left(-i\hbar \dfrac{d}{dx}\right) \, \Psi \, (x,t) \, dx, \label{7.9}$, where the quantity in parentheses, sandwiched between the wavefunctions, is called the momentum operator in the x-direction. This is usually given the Greek letter Ψ (psi) and is a function of position ( x ) and time ( t ), and it contains all of the information that can be known about the particle. He said they were meaningless, because in order to find out you have to conduct a measurement, and the form of the measurement (i.e. Adopted or used LibreTexts for your course? In addition, all measurements are fundamentally probabilistic, and this probability is built into nature rather than being due to a lack of knowledge or precision on the part of the scientists. For a particle in two dimensions, the integration is over an area and requires a double integral; for a particle in three dimensions, the integration is over a volume and requires a triple integral. Later in this section, you will see how to use the wavefunction to describe particles that are “free” or bound by forces to other particles. Finally, the probability density is, |\psi|^2 = (2/L) \, \sin^2 (\pi x/L). A baseball thrown though a window transfers energy from one point The interpretation of $$\Psi^* (x,t) \, \Psi \, (x,t)$$ as a probability density ensures that the predictions of quantum mechanics can be checked in the “real world.”, Suppose that a particle with energy E is moving along the x-axis and is confined in the region between 0 and L. One possible wavefunction is, \[\psi (x,t) =\begin{cases} For example, the wave function spreads across space, and this means that the particle itself doesn’t have a fixed location until you measure it, at which point the wave function “collapses,” and you obtain a definite value. This procedure eliminates complex numbers in all predictions because the product $$\Psi^* (x,t) \, \Psi \, (x,t)$$ is always a real number. He was also a science blogger for Elements Behavioral Health's blog network for five years. The Copenhagen interpretation is the most well-known attempt to answer questions like this and still the most widely-accepted one. [The momentum operator in Equation \ref{7.9} is said to be the position-space representation of the momentum operator.] \nonumber \end{align*}, \begin{align*}\langle x \rangle &= \int_0^L dx \, \psi^* (x) x \psi(x) \nonumber \\[4pt] &= \int_0^L dx \, \left(A e^{+i\omega t} \sin \, \dfrac{\pi x}{L}\right) x \left(A e^{-i\omega t} \sin \, \dfrac{\pi x}{L}\right) \nonumber \\[4pt] &= A^2 \int_0^L dx\,x \, \sin^2 \, \dfrac{\pi x}{L} \nonumber \\[4pt] &= A^2 \dfrac{L^2}{4} \nonumber \\[4pt] \Rightarrow A &= \dfrac{L}{2}. This is opposed to having a “continuous” range of possible values, like quantities at the macro scale. A peculiarity of quantum theory is that these functions are usually complex functions. The ball is equally like to be found anywhere in the box, so one way to describe the ball with a constant wavefunction (Figure $$\PageIndex{3}$$). \nonumber \end{align} \nonumber, Thus, the expectation value of the kinetic energy is, \begin{align*} \langle K \rangle &= \int_0^L dx \left( Ae^{+i\omega t} \, \sin \, \dfrac{\pi x}{L}\right) \left(\dfrac{Ah^2}{8mL^2} e^{-i\omega t} \, \sin \, \dfrac{\pi x}{L}\right) \nonumber \\[4pt] &= \dfrac{A^2h^2}{8mL^2} \int_0^L dx \, \sin^2 \, \dfrac{\pi x}{L} \nonumber \\[4pt] &= \dfrac{A^2h^2}{8mL^2} \dfrac{L}{2} \nonumber \\[4pt] &= \dfrac{h^2}{8mL^2}. If it is necessary to find the probability that a particle will be found in a certain interval, square the wavefunction and integrate over the interval of interest. The square of the matter wave $$|\Psi|^2$$ in one dimension has a similar interpretation as the square of the electric field $$|E|^2$$. The probability of finding the particle “somewhere” (the normalization condition) is, \[P(-\infty, +\infty) = \int_{-\infty}^{\infty} |\Psi \, (x,t)|^2 dx = 1.\label{7.4}. (This is analogous to squaring the electric field strength—which may be positive or negative—to obtain a positive value of intensity.) Note that these conclusions do not depend explicitly on time. In particular, the wavefunction is given by, $\Psi \, (x,t) = A \, \cos \, (kx - \omega t) + i A \, \sin \, (kx - \omega t), \label{eq56}$, where $$A$$ is the amplitude, $$k$$ is the wave number, and $$ω$$ is the angular frequency. The output may be a probability distribution, but it still manages to be complete in its description. In quantum mechanics, though, nature itself sets a limit to the precision you can measure two non-commuting observables to. \nonumber\], Substitute the wavefunction into Equation \ref{7.7} and evaluate. Third, if a matter wave is given by the wavefunction $$\Psi \, (x,t)$$, where exactly is the particle? What precisely is “waving”? The purpose of this chapter is to answer these questions. Wave-particle duality is one of the key concepts in quantum physics, and that’s why each particle is represented by a wave function. Each of these properties is described in more detail below. \end{align*} \], The average position of a large number of particles in this state is $$L/2$$. This function is produced by reflecting $$\psi (x)$$ for $$x > 0$$ about the vertical y-axis. The many worlds interpretation was proposed by Hugh Everett III, and essentially removes the need for the collapse of the wave function entirely, but in doing so proposes multiple parallel “worlds” (which has a slippery definition in the theory) coexisting with your own. In your world, for example, the particle is at position A rather than B or C, but in another world it will be at B, and in yet another it will be at C. This is in essence a deterministic (rather than a probabilistic theory), but it’s your uncertainty about which world you inhabit that creates the apparently probabilistic nature of quantum mechanics. Unlike a digital computer, which encodes information in binary digits (zeroes and ones), a quantum computer stores and manipulates data in the form of quantum bits, or qubits. In this view, the wave-particle duality of quantum mechanics doesn’t mean that a particle is both a wave and a particle; it simply means that a particle like an electron will behave as a wave in some circumstances and as a particle in others. We use the same strategy as before. There is a fundamental limit to the level of accuracy with which you can measure both of these quantities simultaneously. The integral vanishes because the total area of the function about the x-axis cancels the (negative) area below it. Wave-particle duality is one of the key concepts in quantum physics, and that’s why each particle is represented by a wave function. There are many properties that scientists use to describe waves. A simple type of wave is illustrated below. You might not fully understand what exactly is happening – because the behavior of matter at this scale is so weird it almost defies explanation – but the tools scientists have developed to describe quantum theory are indispensable to any physicist. The expectation value of momentum in the x-direction also requires an integral. The modulus (i.e. To determine the probability of finding the ball in the last quarter of the tube, we square the function and integrate: $P(x = L/4, L/2) = \int_{L/4}^{L/2} \left|\sqrt{\dfrac{2}{L}} \, \cos \, \left(\dfrac{\pi x}{L}\right) \right| ^2 dx = 0.091.$, The probability of finding the ball in the last quarter of the tube is 9.1%. However, an odd function times an even function produces an odd function, such as $$x^2e^{-x^2}$$ (odd times even is odd). The Schrodinger equation is the most important equation in quantum mechanics, and it describes the evolution of wave function with time, and allows you to determine the value of it. Period: the time it takes for one complete wave to pass a given point, measured in seconds. The particle has many values of position for any time $$t$$, and only the probability density of finding the particle, $$|\Psi \, (x,t)|^2$$, can be known.