The Particle in a Ring of Constant Potential

Suppose that a particle of mass m is free to move around a ring of radius r and zero potential, but that it requires infinite energy to get off the ring. This system has only one variable coordinate— the angle j). In classical mechanics, the useful quantities and relationships for describing such circular motion are those given in Table 2-1. 

Comparing formulas for linear momentum and angular momentum reveals that the variables mass and linear velocity are analogous to moment of inertia and angular velocity in circular motion, where the coordinate replaces x. The Schrodinger equation for circular motion, then, is 

Angular momentum (linear momentum times orbit radius) mvr = Ico g cm /s or erg s or J s 

Let us solve the problem first with the trigonometric functions. Starting at some arbitrary point on the ring and moving around the circumference with a sinusoidal function, we shall eventually reencounter the initial point. In order that our wavefunction be single valued, it is necessary that repeat itself every time / changes by In radians. Thus, for 4 given by Eq. (2-45), 

Either of these relations is satisfied only if is an integer. The case in which = 0 is not allowed for the sine function since it then vanishes everywhere and is unsuitable. However, A = 0 is allowed for the cosine form. The normalized solutions are, then. 

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Section 2-6 The Particle in a Ring of Constant Potential TABLE 2-1 51... 

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