1. Suppose that an electron is trapped in a one-dimensional, infinite potential

Question

1. Suppose that an electron is trapped in a one-dimensional, infinite potential well of width 250 nm is excited from the 2nd excited state to the fifth excited state. What energy must be transferred to the electron in order to make this transition?

2. What wavelength photon does this correspond to?

3. Considering all of the possible ways that the excited electron can de-excite back down to the ground state, what is the longest wavelength photon that could be emitted?

4. Considering all of the possible ways the electron could de-excite back down to the ground state, what is the shortest wavelength photon that could be emitted?

5. What is the wavelength of an electron in the ground state of a hydrogen atom?

6. A beam of electrons is incident upon a gas of hydrogen atoms. What minimum speed must the electrons have to cause the emission of 486 nm light corresponding to the 4 to 2 transitions of hydrogen? Report your answer to 3 significant figures.

Please help by answering the question 3, question 4 and question 6 step by step. This is college physics, calculus based. Please answer all parts thank you.

1. Suppose that an electron is trapped in a one-dimensional, infinite potential well of width 250 nm is excited from the 2nd excited state to the fifth excited state What energy must be transferred to the electron in order to make this transition? 2. What wavelength photon does this correspond to? 3. Considering all of the possible ways that the excited electron can de-excite back down to the ground state, what is the longest wavelength photon that could be emitted? 4. Considering all of the possible ways the electron could de- excite back down to the ground state, what is the shortest wavelength photon that could be emitted? 5. What is the wavelength of an electron in the ground state of a hydrogen atom? 6. A beam of electrons is incident upon a gas of hydrogen atoms What minimum speed must the electrons have to cause the emission of 486 nm light corresponding to the 4 to 2 transitions of hydrogen? Report your answer to 3 significant figures

Explanation / Answer

1. En = n^2 h^2 / (8 m L^2)

for 2nd excited state, n = 3

for fifth excited state, n = 6

delta (E) = (6^2 - 3^2)(6.626 x 10^-34)^2 / (8 x 9.11 x 10^-31 x (250 x 10^-9)^2)

= 2.602 x 10^-23 J ........Ans

2. wavelength = h c / E

= (6.662 x 10^-34)(3 x 10^8) / (2.602 x 10^-24)

= 7.64 x 10^-3 m OR 7.64 mm ....Ans

3. longest wavelength when E is smallest.

(n -> 2 to 1)

E= (2^2 - 1^2) h^2 / (8 m L^2) = 2.891 x 10^-24 J

wavelength = h c / E = 0.0688 m or 68.8 mm  

4. Shortest wavelength when delta(E) will be max.

E = (6^2 - 1^2)(h^2) / (8 m L^2) = 3.373 x 10^-23 J  

wavelength = 5.32 x 10^-3 m Or 5.32 mm

5. for ground state,

wavelength = 2 L = 5 x 10^-7 m OR 500 nm

6. p = h / lambda = (6.626 x 10^-34) /(486 x 10^-9)

p = 1.3634 x 10^-27 kg m/s

p = m v

v = 1500 m/s

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