chapter5 - Physics 107 Problem 5.1 Which of the wave...

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(f) YES. The function in figure 5.16(f) appears to satisfy all of the conditions, so it can be a wave function, although it is not clear how the amplitude can decrease with increasing x without the wavelength changing. (e) NO. The function is not single valued. (d) YES. (c) NO. The first derivative is discontinuous in the middle. Beiser also probably intends to imply that the function goes to infinity at the middle. (b) YES if you assume the "interval shown" is only where the function is defined. NO if you assume the "interval shown" means the entire shown part of the positive x-axis. (a) YES. "YES" means allowed, "NO" means not allowed. Which of the wave functions in Fig. 5.16 cannot have physical significance in the interval shown? Why not? Problem 5.2 Physics 107 The function in figure 5.15(f) is not continuous, so it cannot be a wave function. This is tricky, because it is possible that the derivative could be continuous and finite. The function in figure 5.15(e) appears to satisfy all of the conditions, so it can be a wave function. The function in figure 5.15(d) does not satisfy the condition for a continuous first derivative, so it cannot be a wave function. In addition, the function is not continuous. Beiser in his answer implies the function goes to infinity, which is not obvious from the figure as it is drawn. The function in figure 5.15(c) does not satisfy the condition for a continuous first derivative, so it cannot be a wave function. The function in figure 5.15(b) is not single-valued, so it cannot be a wave function. The function in figure 5.15(a) appears to satisfy all of the conditions over the interval shown, so it can be a wave function. If you want to be picky, its zero slope everywhere means the momentum is zero everywhere, which is not allowed by the uncertainty principle. I will not be that tricky on the exam. must be finite, and that Φ must be single-valued and continuous with finite, single-valued, and continuous first derivatives. V Φ () 2 d Recall that Which of the wave functions in Fig. 5.15 cannot have physical significance in the interval shown? Why not? Problem 5.1 Physics 107 1
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This function has singularities at π /2, 3 π /2, etc. . Φ x () x x2 −π 2 2 π 100 + , 2 π .. := Φ x At a nx := (b) There are singularities wherever cos(x)=0, so this is not an allowed wave function if its range includes an x-value where there is a singularity. Φ x x We can see the function better if we choose the plot scale ourselves. There are singularities here, which seem to scale differently. Because of our choice of values of x, the function does not always "hit" the singularity at exactly the same position. Φ x x First plot Φ (x), letting Mathcad choose the plot scale. 2 2 π 100 + , 2 π .. := Φ x 1 cos x := (a) The secant is 1/cosine. Let's plot the secant function for small values of x.
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chapter5 - Physics 107 Problem 5.1 Which of the wave...

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