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Energy of Wave

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For a particle of mass m enclosed in a one - dimensional box of length L, the de - Broglie concept would lead to stationary waves, with nodes at the two ends, The energy value allowed for such a system will be?

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Q. When a particle is restricted to move along the x-axis between x = 0 and x = a, where 'a' is of nanometer dimension, its energy can take only certain specific values. The allowed energies of the particle moving in such a restricted region, correspond to the formation of standing waves with nodes at its ends x = 0 and x = a. The wavelength of this standing wave is related to the linear momentum p of the particle according to the de Broglie relation. The energy of the particle of mass, 'm' is related to its linear momentum as

E = p^{2} / 2 m .

Thus, the energy of the particle can be denoted by a quantum number 'n' taking values 1,2,3,.... (n=1, called the ground state) corresponding to the number of loops in the standing wave.
Take

h = 6.6 \times 10^{- 34} J s a n d e = 1.6 \times 10^{- 19} C .

The allowed energy for the particle for a particular value of n is proportional to

Q. Assume that the de Broglie wave associated with an electron can form a standing wave between the atoms arranged in a one dimensional array with nodes at each of the atomic sites. It is found that one such standing wave is formed if the distance

^{'} d^{'}

between the atoms of the array is

2 ˚ A

. A similar standing wave is again formed if

^{'} d^{'}

is increased to

2.5 ˚ A

but not for any intermediate value of

d

.
Find the energy of the electrons in

e V

:

Q. When a particle is restricted to move along the x-axis between x = 0 and x = a, where 'a' is of nanometer dimension, its energy can take only certain specific values. The allowed energies of the particle moving in such a restricted region, correspond to the formation of standing waves with nodes at its ends x = 0 and x = a. The wavelength of this standing wave is related to the linear momentum p of the particle according to the de Broglie relation. The energy of the particle of mass, 'm' is related to its linear momentum as

E = p^{2} / 2 m .

Thus, the energy of the particle can be denoted by a quantum number 'n' taking values 1,2,3,.... (n=1, called the ground state) corresponding to the number of loops in the standing wave.
Take

h = 6.6 \times 10^{- 34} J s a n d e = 1.6 \times 10^{- 19} C .

If the mass of the particle is

m = 1.0 \times 10^{- 30} k g

and radius, a = 6.6 nm, the energy of the particle in its ground state is closest to

Q. Assume that the de-Broglie wave associated with an electron can form a standing wave between the atoms arranged in a one dimensional array with nodes at each of the atomic sites. It is found that one such standing wave is formed if the distance

^{'} d^{'}

between the atoms of the array is

2 A^{\circ}

. A similar standing wave is again formed if

^{'} d^{'}

is increased to

2.5 A^{\circ}

but not for any intermediate value of

^{'} d^{'}

. Find the energy of the electron in electron volts and the least value of

^{'} d^{'}

for which the standing wave of the type described above can form.

Q. Assume that the de-Broglie wave associated with an electron can form a standing wave between the atoms arranged in a one dimensional array with nodes at each of the atomic sites. It is found that one such standing wave is formed if the distance

d

between the atoms of the array is

2 ˚ A

. A similar standing wave is again formed if

d

is increased to

2.5 ˚ A

but not for any intermediate value of

d

. Find the least value of

d

for which the standing wave of the type described above can form.

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