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

where Ψ * ( x , t ) is the complex conjugate of the wave function. The complex conjugate of a function is obtaining by replacing every occurrence of i = −1 in that function with i . This procedure eliminates complex numbers in all predictions because the product Ψ * ( x , t ) Ψ ( x , t ) is always a real number.

Check Your Understanding If a = 3 + 4 i , what is the product a * a ?

( 3 + 4 i ) ( 3 4 i ) = 9 16 i 2 = 25

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Consider the motion of a free particle that moves along the x -direction. As the name suggests, a free particle experiences no forces and so moves with a constant velocity. As we will see in a later section of this chapter, a formal quantum mechanical treatment of a free particle indicates that its wave function has real and complex parts. In particular, the wave function is given by

Ψ ( x , t ) = A cos ( k x ω t ) + i A sin ( k x ω t ) ,

where A is the amplitude, k is the wave number, and ω is the angular frequency. Using Euler’s formula, e i ϕ = cos ( ϕ ) + i sin ( ϕ ) , this equation can be written in the form

Ψ ( x , t ) = A e i ( k x ω t ) = A e i ϕ ,

where ϕ is the phase angle. If the wave function varies slowly over the interval Δ x , the probability of finding the particle in that interval is

P ( x , x + Δ x ) Ψ * ( x , t ) Ψ ( x , t ) Δ x = ( A e i ϕ ) ( A * e i ϕ ) Δ x = ( A * A ) Δ x .

If A has real and complex parts ( a + i b , where a and b are real constants), then

A * A = ( a + i b ) ( a i b ) = a 2 + b 2 .

Notice that the complex numbers have vanished. Thus,

P ( x , x + Δ x ) | A | 2 Δ x

is a real quantity. The interpretation of Ψ * ( x , t ) Ψ ( x , t ) as a probability density ensures that the predictions of quantum mechanics can be checked in the “real world.”

Check Your Understanding 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 wave function is

ψ ( x , t ) = { A e i E t / sin π x L , when 0 x L 0 , otherwise .

Determine the normalization constant.

A = 2 / L

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Expectation values

In classical mechanics, the solution to an equation of motion is a function of a measurable quantity, such as x ( t ), where x is the position and t is the time. Note that the particle has one value of position for any time t . In quantum mechanics, however, the solution to an equation of motion is a wave function, Ψ ( x , t ) . The particle has many values of position for any time t , and only the probability density of finding the particle, | Ψ ( x , t ) | 2 , can be known. The average value of position for a large number of particles with the same wave function is expected to be

x = x P ( x , t ) d x = x Ψ * ( x , t ) Ψ ( x , t ) d x .

This is called the expectation value    of the position. It is usually written

x = Ψ * ( x , t ) x Ψ ( x , t ) d x ,

where the x is sandwiched between the wave functions. The reason for this will become apparent soon. Formally, x is called the position operator    .

At this point, it is important to stress that a wave function can be written in terms of other quantities as well, such as velocity ( v ), momentum ( p ), and kinetic energy ( K ). The expectation value of momentum, for example, can be written

p = Ψ * ( p , t ) p Ψ ( p , t ) d p ,

Where dp is used instead of dx to indicate an infinitesimal interval in momentum. In some cases, we know the wave function in position, Ψ ( x , t ) , but seek the expectation of momentum. The procedure for doing this is

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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