When a particle is restricted to move along x- axis between `x = 0 and x = 4 ` whwre a is opf nanometer demension , its energy can take only certain spscfic values . The allowed energies of the particles only in such a restiricted regain , correspond to the formation of standing wave with nodes at its end ` x = 0 and x = a `.The wavelength of this standing wave is related to the linear momentum p of the paarticle according to the de Broglie relation .The energy of the particle of mass `m` is reated to its linear momentum as
`E = (p^(2))/(2m)` . thus , the energy of the particle can be denoted by a quantum number `n` taking value `1,``2,``3,``...``(n= 1,` called the ground state) corresponding to the number of loops in the standing wave use the model described above to answer the following there question for a particle moving in the line ` x = 0` to `x = a` Take `h = 6.6 xx 10^(-34)` J s and `e = 1.6 xx 10^(-19)C`
The allowed energy for the particle for a particular value of `n` is proportional to
When a particle is restricted to move along x- axis between `x = 0 and x = 4 ` whwre a is opf nanometer demension , its energy can take only certain spscfic values . The allowed energies of the particles only in such a restiricted regain , correspond to the formation of standing wave with nodes at its end ` x = 0 and x = a `.The wavelength of this standing wave is related to the linear momentum p of the paarticle according to the de Broglie relation .The energy of the particle of mass `m` is reated to its linear momentum as
`E = (p^(2))/(2m)` . thus , the energy of the particle can be denoted by a quantum number `n` taking value `1,``2,``3,``...``(n= 1,` called the ground state) corresponding to the number of loops in the standing wave use the model described above to answer the following there question for a particle moving in the line ` x = 0` to `x = a` Take `h = 6.6 xx 10^(-34)` J s and `e = 1.6 xx 10^(-19)C`
The allowed energy for the particle for a particular value of `n` is proportional to
`E = (p^(2))/(2m)` . thus , the energy of the particle can be denoted by a quantum number `n` taking value `1,``2,``3,``...``(n= 1,` called the ground state) corresponding to the number of loops in the standing wave use the model described above to answer the following there question for a particle moving in the line ` x = 0` to `x = a` Take `h = 6.6 xx 10^(-34)` J s and `e = 1.6 xx 10^(-19)C`
The allowed energy for the particle for a particular value of `n` is proportional to
A
`a^(-2)`
B
`a^(-3//2)`
C
`a^(-1)`
D
`a^(2)`
Text Solution
Verified by Experts
The correct Answer is:
A
` lambda = (h)/(p) and E = (p^(2))/(2 m)`
rArr E = (h^(2))/(2m lambda^(2))`
The length in which the particle is restricted is move is a This length is a multiple of `(lambda)/(2)`
Now , `n(lambda)/(2) = u rArr lambda = (2a)/(n)`
`rArr E = ((h^(2) n^(2))/(2 m xx 4a^(2)) = (n^(2) h^(2))/(8 ma^(2))` rArr E prop a^(-2)` for a particular value of `n`
rArr E = (h^(2))/(2m lambda^(2))`
The length in which the particle is restricted is move is a This length is a multiple of `(lambda)/(2)`
Now , `n(lambda)/(2) = u rArr lambda = (2a)/(n)`
`rArr E = ((h^(2) n^(2))/(2 m xx 4a^(2)) = (n^(2) h^(2))/(8 ma^(2))` rArr E prop a^(-2)` for a particular value of `n`
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