Fund Quantum Mechanics Lect & HW Solutions 29

Fund Quantum Mechanics Lect & HW Solutions 29 - of...

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2.4. OPERATORS 11 2.4.1 Solution mathops-a Question: So what is the result if the operator d / d x is applied to the function sin( x )? Answer: Its derivative, the function cos( x ), [1, p. 60]. 2.4.2 Solution mathops-b Question: If, say, h x 2 sin( x ) is simply the function x 2 sin( x ), then what is the diFerence between h x 2 and x 2 ? Answer: Nothing that aFects the price of eggs. Just the way you think about them. You think of h x 2 as the operator that turns a function like sin( x ) into the function x 2 sin( x ). But you think of x 2 as a recipe that turns a value like 3 into 3 2 = 9. 2.4.3 Solution mathops-c Question: A less self-evident operator than the above examples is a translation operator like T π/ 2 that translates the graph of a function towards the left by an amount π/ 2: T π/ 2 f ( x ) = f p x + 1 2 π P . (Curiously enough, translation operators turn out to be responsible for the law
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Unformatted text preview: of conservation of momentum.) Show that T / 2 turns sin( x ) into cos( x ). Answer: Using various standard trig manipulations, [1, pp. 43-44]: T / 2 sin( x ) = sin p x + 1 2 P = cos( x ) = cos( x ) . Or just compare the graphs visually, [1, p. 43]. 2.4.4 Solution mathops-d Question: The inversion operator Inv turns f ( x ) into f ( x ). It plays a part in the question to what extent physics looks the same when seen in the mirror. Show that Inv leaves cos( x ) unchanged, but turns sin( x ) into sin( x ). Answer: According to [1, p. 43], cos( x ) = cos( x ), but sin( x ) = sin( x ). Compare also the graphs of the functions on the same page; Inv ips the graph of a function over around the y-axis....
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