The wave length of a quantum mechanical particle in a potential well U ( x ) as being like a free particle as a function of position x can be written as λ ( x ) = h 2 m [ E − U ( x ) ] .

Question

Hi! I need help with these calculations for part i and part k for a physics Diffraction Lab. We used a slit width 0.4 mm to measure our pattern.

Answer 1

Question

Chapter 40, Problem 40.64CP

(a)

To determine

To show: The wave length of a quantum mechanical particle in a potential well $U(x)$ as being like a free particle as a function of position $x$ can be written as $λ(x)=h2m[E−U(x)]$ .

(b)

To determine

What happens to its wavelength if the particle moves into the region of increasing potential energy.

(b)

To determine

What happens to its wavelength if the particle moves into the region of increasing potential energy.

(c)

To determine

The wavelength at classical turning point.

(d)

To determine

To show: That the WKB condition for an allowed bound state energy can be written as $∫ab2m[E−U(x)]dx=nh2$ for $n=1,2,3,....$ .

(e)

To determine

To show: That Energy for particle in a box with infinite potential walls at and $x=L$ using WKB approximation is equal to $n2h28mL2$ , where $n=1,2,..$ .

(f)

To determine

The integral $∫ab2m[E−U(x)]dx=nh2$ for a finite potential well with walls at $x=0$ and $x=L$ then show that allowed energy levels according to the WKB approximation are the same as those obtained in infinite potential well.

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Answer 2

Question

Chapter 40, Problem 40.64CP

(a)

To determine

To show: The wave length of a quantum mechanical particle in a potential well $U(x)$ as being like a free particle as a function of position $x$ can be written as $λ(x)=h2m[E−U(x)]$ .

(b)

To determine

What happens to its wavelength if the particle moves into the region of increasing potential energy.

(b)

To determine

What happens to its wavelength if the particle moves into the region of increasing potential energy.

(c)

To determine

The wavelength at classical turning point.

(d)

To determine

To show: That the WKB condition for an allowed bound state energy can be written as $∫ab2m[E−U(x)]dx=nh2$ for $n=1,2,3,....$ .

(e)

To determine

To show: That Energy for particle in a box with infinite potential walls at and $x=L$ using WKB approximation is equal to $n2h28mL2$ , where $n=1,2,..$ .

(f)

To determine

The integral $∫ab2m[E−U(x)]dx=nh2$ for a finite potential well with walls at $x=0$ and $x=L$ then show that allowed energy levels according to the WKB approximation are the same as those obtained in infinite potential well.

Expert Solution & Answer

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Answer 3

Question

Chapter 40, Problem 40.64CP

(a)

To determine

To show: The wave length of a quantum mechanical particle in a potential well $U(x)$ as being like a free particle as a function of position $x$ can be written as $λ(x)=h2m[E−U(x)]$ .

(b)

To determine

What happens to its wavelength if the particle moves into the region of increasing potential energy.

(b)

To determine

What happens to its wavelength if the particle moves into the region of increasing potential energy.

(c)

To determine

The wavelength at classical turning point.

(d)

To determine

To show: That the WKB condition for an allowed bound state energy can be written as $∫ab2m[E−U(x)]dx=nh2$ for $n=1,2,3,....$ .

(e)

To determine

To show: That Energy for particle in a box with infinite potential walls at and $x=L$ using WKB approximation is equal to $n2h28mL2$ , where $n=1,2,..$ .

(f)

To determine

The integral $∫ab2m[E−U(x)]dx=nh2$ for a finite potential well with walls at $x=0$ and $x=L$ then show that allowed energy levels according to the WKB approximation are the same as those obtained in infinite potential well.

Expert Solution & Answer

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Answer 4

Question

Chapter 40, Problem 40.64CP

(a)

To determine

To show: The wave length of a quantum mechanical particle in a potential well $U(x)$ as being like a free particle as a function of position $x$ can be written as $λ(x)=h2m[E−U(x)]$ .

(b)

To determine

What happens to its wavelength if the particle moves into the region of increasing potential energy.

(b)

To determine

What happens to its wavelength if the particle moves into the region of increasing potential energy.

(c)

To determine

The wavelength at classical turning point.

(d)

To determine

To show: That the WKB condition for an allowed bound state energy can be written as $∫ab2m[E−U(x)]dx=nh2$ for $n=1,2,3,....$ .

(e)

To determine

To show: That Energy for particle in a box with infinite potential walls at and $x=L$ using WKB approximation is equal to $n2h28mL2$ , where $n=1,2,..$ .

(f)

To determine

The integral $∫ab2m[E−U(x)]dx=nh2$ for a finite potential well with walls at $x=0$ and $x=L$ then show that allowed energy levels according to the WKB approximation are the same as those obtained in infinite potential well.

Expert Solution & Answer

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Answer 5

Chapter 40, Problem 40.64CP

(a)

To determine

To show: The wave length of a quantum mechanical particle in a potential well $U(x)$ as being like a free particle as a function of position $x$ can be written as $λ(x)=h2m[E−U(x)]$ .

(b)

To determine

What happens to its wavelength if the particle moves into the region of increasing potential energy.

(b)

To determine

What happens to its wavelength if the particle moves into the region of increasing potential energy.

(c)

To determine

The wavelength at classical turning point.

(d)

To determine

To show: That the WKB condition for an allowed bound state energy can be written as $∫ab2m[E−U(x)]dx=nh2$ for $n=1,2,3,....$ .

(e)

To determine

To show: That Energy for particle in a box with infinite potential walls at and $x=L$ using WKB approximation is equal to $n2h28mL2$ , where $n=1,2,..$ .

(f)

To determine

The integral $∫ab2m[E−U(x)]dx=nh2$ for a finite potential well with walls at $x=0$ and $x=L$ then show that allowed energy levels according to the WKB approximation are the same as those obtained in infinite potential well.

Answer 6

Chapter 40, Problem 40.64CP

(a)

To determine

To show: The wave length of a quantum mechanical particle in a potential well $U(x)$ as being like a free particle as a function of position $x$ can be written as $λ(x)=h2m[E−U(x)]$ .

Answer 7

Chapter 40, Problem 40.64CP

(a)

To determine

To show: The wave length of a quantum mechanical particle in a potential well $U(x)$ as being like a free particle as a function of position $x$ can be written as $λ(x)=h2m[E−U(x)]$ .

Answer 8

(b)

To determine

What happens to its wavelength if the particle moves into the region of increasing potential energy.

Answer 9

(b)

To determine

What happens to its wavelength if the particle moves into the region of increasing potential energy.

Answer 10

(b)

To determine

What happens to its wavelength if the particle moves into the region of increasing potential energy.

Answer 11

(b)

To determine

What happens to its wavelength if the particle moves into the region of increasing potential energy.

Answer 12

(c)

To determine

The wavelength at classical turning point.

Answer 13

(c)

To determine

The wavelength at classical turning point.

Answer 14

(d)

To determine

To show: That the WKB condition for an allowed bound state energy can be written as $∫ab2m[E−U(x)]dx=nh2$ for $n=1,2,3,....$ .

Answer 15

(d)

To determine

To show: That the WKB condition for an allowed bound state energy can be written as $∫ab2m[E−U(x)]dx=nh2$ for $n=1,2,3,....$ .

Answer 16

(e)

To determine

To show: That Energy for particle in a box with infinite potential walls at and $x=L$ using WKB approximation is equal to $n2h28mL2$ , where $n=1,2,..$ .

Answer 17

(e)

To determine

To show: That Energy for particle in a box with infinite potential walls at and $x=L$ using WKB approximation is equal to $n2h28mL2$ , where $n=1,2,..$ .

Answer 18

(f)

To determine

The integral $∫ab2m[E−U(x)]dx=nh2$ for a finite potential well with walls at $x=0$ and $x=L$ then show that allowed energy levels according to the WKB approximation are the same as those obtained in infinite potential well.

Answer 19

(f)

To determine

The integral $∫ab2m[E−U(x)]dx=nh2$ for a finite potential well with walls at $x=0$ and $x=L$ then show that allowed energy levels according to the WKB approximation are the same as those obtained in infinite potential well.

Answer 20

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Ch. 40.1 - Does a wave packet given by Eq. (40.19) represent...Ch. 40.2 - Prob. 40.2TYU Ch. 40.3 - Prob. 40.3TYU Ch. 40.4 - Prob. 40.4TYU Ch. 40.5 - Prob. 40.5TYU Ch. 40.6 - Prob. 40.6TYU Ch. 40 - Prob. 40.1DQ Ch. 40 - Prob. 40.2DQ Ch. 40 - Prob. 40.3DQ Ch. 40 - Prob. 40.4DQ

The wave length of a quantum mechanical particle in a potential well U ( x ) as being like a free particle as a function of position x can be written as λ ( x ) = h 2 m [ E − U ( x ) ] .

Solution Summary: The author explains how the wave length of a quantum mechanical particle in the potential well U(x) can be written as lambda

To show: The wave length of a quantum mechanical particle in a potential well $U(x)$ as being like a free particle as a function of position $x$ can be written as $λ(x)=h2m[E−U(x)]$ .

What happens to its wavelength if the particle moves into the region of increasing potential energy.

What happens to its wavelength if the particle moves into the region of increasing potential energy.

The wavelength at classical turning point.

To show: That the WKB condition for an allowed bound state energy can be written as $∫ab2m[E−U(x)]dx=nh2$ for $n=1,2,3,....$ .

To show: That Energy for particle in a box with infinite potential walls at and $x=L$ using WKB approximation is equal to $n2h28mL2$ , where $n=1,2,..$ .

The integral $∫ab2m[E−U(x)]dx=nh2$ for a finite potential well with walls at $x=0$ and $x=L$ then show that allowed energy levels according to the WKB approximation are the same as those obtained in infinite potential well.

Chapter 40 Solutions