Physical interpretation of the wave function

Young s double-slit experiment and the Stem-Gerlach experiment, as described in the two previous sections, lead to a physical interpretation of the wave function associated with the motion of a particle. Basic to the concept of the wave function is the postulate that the wave function contains all the 

2 This point is discussed in more detail in N. F. Mott and H. S. W. Massey (1965) The Theory of Atomic Collisions, 3rd edition, p. 215-16, (Oxford University Press, Oxford). 

If the motion of a particle in the double-slit experiment is to be represented by a wave function, then that wave function must determine the probability density P(x). For mechanical waves in matter and for electromagnetic waves, the intensity of a wave is proportional to the square of its amplitude. By analogy, the probability density P(x) is postulated to be the square of the absolute value of the wave function ftftx) 

On the basis of this postulate, the interference pattern observed in the doubleslit experiment can be explained in terms of quantum particle behavior. 

A particle, photon or electron, passing through slit A and striking the detection screen at point x has wave function a(x), while a similar particle passing through slit B has wave function b(x). Since a particle is observed to retain its identity and not divide into smaller units, its wave function Fix) is postulated to be the sum of the two possibilities 

When only slit A is open, the particle emitted by the source S passes through slit A, thereby causing the wave function ft (x) in equation (1.48) to change or collapse suddenly to ft A( )- The probability density 7a(x) that the particle strikes point x on the detection screen is, then 

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Our presentation of the basic principles of quantum mechanics is contained in the first three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle. Several experiments, which lead to a physical interpretation of the wave function, are also described. In Chapter 2, the Schrodinger differential wave equation is introduced and the wave function concept is extended to include particles in an external potential field. The formal mathematical postulates of quantum theory are presented in Chapter 3. 

Another experiment that relates to the physical interpretation of the wave function was performed by O. Stem and W. Gerlach (1922). Their experiment is a dramatic illustration of a quantum-mechanical effect which is in direct conflict with the concepts of classical theory. It was the first experiment of a non-optical nature to show quantum behavior directly. 

The wave function for this system is a function of the N position vectors (ri, r2,. .., r v, i). Thus, although the N particles are moving in three-dimensional space, the wave function is 3iV-dimensional. The physical interpretation of the wave function is analogous to that for the three-dimensional case. The quantity... 

A time-independent wave function is a function of the position in space (r = x,y,z) and the spin degree of freedom, which can be either up or down. The physical interpretation of the wave function is due to Max Born (25, 26), who was the first to interpret the square of its magnitude, > /(r)p, as a probability density function, or probability distribution function. This probability distribution specifies the probability of finding the particle (here, the electron) at any chosen location in space (r) in an infinitesimal volume dV= dx dy dz around r. I lu probability of finding the electron at r is given by )/(r) Id V7, which is required to integrate to unity over all space (normalization condition). A many-electron system, such as a molecule, is described by a many-electron wave function lF(r, r, l .I -.-), which when squared gives the probability den-... 

There needs to be some physical interpretation of the wave function and its relationship to the state of the system. One interpretation is that the square of the wave function, ip2, is proportional to the probability of finding the parts of the system in a specified region of space. For some problems in quantum mechanics, differential equations arise that can have solutions that are complex (contain (-l)1/2 = i). In such a case, we use ip ip, where ip is the complex conjugate of ip. The complex conjugate of a function is the function that results when i is replaced by — i. Suppose we square the function (a + ib) ... 

The physical interpretation of the wave function of a particle is that, its square at a particular point is the probability density of finding the particle there. ... 

It will be shown later that the general postulates which we shall make regarding the physical interpretation of the wave function require that the constant Wn represent the energy of the system in its various stationary states. 

The wave equation, the auxiliary conditions imposed on the wave functions, and the physical interpretation of the wave functions for the general system are closely similar to those for... 

The physical interpretation of the wave functions for this general system is closely analogous to that for the one-dimensional system discussed in Section 10. We first make the following postulate, generalizing that of Section 10a ... 

In discussing the physical interpretation of the wave functions for this system, let us first consider that the physical situation is represented by a wave function as given in Equation 13-11 with Wy and W, equal to zero and Wx equal to W. The func-... 

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