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Quantum Mechanics

The document contains various problems and solutions related to quantum mechanics, including calculations of de Broglie wavelengths, kinetic energies, uncertainties in position and momentum, and energy levels in quantum systems. It covers topics such as the uncertainty principle, energy states of particles in confined spaces, and the behavior of electrons and protons under different conditions. Each problem is accompanied by its respective answer, providing a comprehensive overview of key concepts in quantum mechanics.

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17 views2 pages

Quantum Mechanics

The document contains various problems and solutions related to quantum mechanics, including calculations of de Broglie wavelengths, kinetic energies, uncertainties in position and momentum, and energy levels in quantum systems. It covers topics such as the uncertainty principle, energy states of particles in confined spaces, and the behavior of electrons and protons under different conditions. Each problem is accompanied by its respective answer, providing a comprehensive overview of key concepts in quantum mechanics.

Uploaded by

veyib
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Quantum Mechanics

1. Find the de Broglie wavelength of (a) 46 g golf ball with velocity 30 m/s and (b) an electron with
velocity 107 m/s. [(a) 4.8 × 10-34 m, (b) 73Ǻ]

2. If the kinetic energy of an electron is 54 eV, what is the wavelength associated with it?
[1.66Ǻ]

3. A measurement establishes the position of a proton with an accuracy of ±10Ǻ. Find the
uncertainty in the proton’s position 1 sec later. Assume the velocity of proton is much smaller
compared to c. [∆𝑥 ≥ 3.15 𝑘𝑚]

4. The radius of a hydrogen atom is 5.3  10 11 m . Use the uncertainty principle to estimate the
minimum energy an electron can have in this atom. Compare your result with lowest energy level
of the hydrogen atom. [𝐾𝐸𝑚𝑖𝑛 = 3.375 𝑒𝑉, 𝐸0 = 13.6 𝑒𝑉]

5. Calculate the de Broglie wavelength of an electron accelerated by the potential difference of


150 V. [1 Ǻ]

6. The position of an electron is located within a distance of 0.1 Ǻ. What is the uncertainty in
measuring the momentum of the electron?
[0.527 × 10−23 𝑘𝑔 𝑚/𝑠]

7. An electron is in a one-dimensional box of 0.1 nm, which is of the order of magnitude of atomic
dimensions. Find the permitted energies. [37.5 n2 eV]

8. A 10 g of marble is in a box 10cm across. Find its permitted energies. [5.5 × 10−64 𝑛2 ]

9. A proton in a 1D box has the energy of 400 keV in its first excited state. How wide the box is?
[45.32 ×10-15 m]

10. The position and momentum of a 1 keV electron are simultaneously determined. If its position is
located within 0.1nm, what is the percentage of minimum uncertainty in its momentum? [3.1%]

11. The de Broglie wavelength of a particle moving with 10  of the velocity of light and that of
proton moving with 20  of the velocity of light are equal. Calculate the wavelength and mass of
the particle. [6.6 × 10-15 m, 3.34 × 10-27 Kg]

12. An electron is confined to move between two rigid walls separated by 10 9 m . Find the de Broglie
wavelengths representing first three allowed energy states and the corresponding energies.
[2,1,2/3 nm ; 0.4,1.5,3.4 eV]
13. If a 15 g of marble moving with a speed of 1/3 ms-1 is confined over a distance of 12 cm, find the
total number of energy levels. What do you infer from the result? [1.81 × 1030]

14. If   A exp( kx) for 0  x   and   0 for    x  0 , find A in terms of k and evaluate the
2 3
probability of the particle lying in the region  x  . [ 2k , 0.016]
k k

15. Find the average momentum of a particle, confined to a one dimensional box of length L, in
ground state. [0]
*****

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