Plane waves and wave packets. In class, we solved the Schrodinger equation for a "free particle" (e.g. when U(x,t) = 0). The correct[solution is (x, t) = Aei(px-Et)/ħ This represents a "plane wave" that exists for all x. However, there is a strange problem with this: if you try to normalize the wave function (determine A by integrating * for all x), you will find an inconsistency (A has to be set equal to 0?). This is because the plane wave stretches to infinity. In order to actually represent a free particle, this solution needs to be handled carefully. Explain in words (and/or diagrams) how we can construct a "wave packet" from the plane wave solution. (Hint 1: consider a superposition of plane waves for a limited range of momentum/energy. Hint 2: have a look at the brief discussion in the middle of pg. 278 and especially pg. 308-309 of the text.)
Plane waves and wave packets. In class, we solved the Schrodinger equation for a "free particle" (e.g. when U(x,t) = 0). The correct[solution is (x, t) = Aei(px-Et)/ħ This represents a "plane wave" that exists for all x. However, there is a strange problem with this: if you try to normalize the wave function (determine A by integrating * for all x), you will find an inconsistency (A has to be set equal to 0?). This is because the plane wave stretches to infinity. In order to actually represent a free particle, this solution needs to be handled carefully. Explain in words (and/or diagrams) how we can construct a "wave packet" from the plane wave solution. (Hint 1: consider a superposition of plane waves for a limited range of momentum/energy. Hint 2: have a look at the brief discussion in the middle of pg. 278 and especially pg. 308-309 of the text.)
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![3. Plane waves and wave packets. In class, we solved the Schrodinger equation for a
"free particle" (e.g. when U(x,t) = 0). The correct[solution is
(x, t) = Ae(px-Et)/ħ
This represents a "plane wave" that exists for all x. However, there is a strange problem
with this: if you try to normalize the wave function (determine A by integrating * for
all x), you will find an inconsistency (A has to be set equal to 0?). This is because the
plane wave stretches to infinity. In order to actually represent a free particle, this solution
needs to be handled carefully. Explain in words (and/or diagrams) how we can construct
a "wave packet" from the plane wave solution. (Hint 1: consider a superposition of plane
waves for a limited range of momentum/energy. Hint 2: have a look at the brief
discussion in the middle of pg. 278 and especially pg. 308-309 of the text.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7565d6cc-3874-4bdb-8336-03f837d820a0%2Ff16b2e4f-dd7e-4c65-a820-2c7b694681df%2Fxo59efy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. Plane waves and wave packets. In class, we solved the Schrodinger equation for a
"free particle" (e.g. when U(x,t) = 0). The correct[solution is
(x, t) = Ae(px-Et)/ħ
This represents a "plane wave" that exists for all x. However, there is a strange problem
with this: if you try to normalize the wave function (determine A by integrating * for
all x), you will find an inconsistency (A has to be set equal to 0?). This is because the
plane wave stretches to infinity. In order to actually represent a free particle, this solution
needs to be handled carefully. Explain in words (and/or diagrams) how we can construct
a "wave packet" from the plane wave solution. (Hint 1: consider a superposition of plane
waves for a limited range of momentum/energy. Hint 2: have a look at the brief
discussion in the middle of pg. 278 and especially pg. 308-309 of the text.)
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