When a particle is restricted to move along x-axis between `x=0` and `x=a`, where `alpha` if of nenometer 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)/(2m)`. 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. Use the model described above to answer the following three questions for a particle moving along the line from `x=0` to `x=alpha`. Take `h=6.6xx10^(-34)Js` and `e=1.6xx10^(-19)`C Q. The speed of the particle that can take discrete values is proportional to
A
`n^((-3)/(2))`
B
`n^(-1)`
C
`n^((1)/(2))`
D
n
Text Solution
AI Generated Solution
To solve the problem, we need to analyze the situation of a particle constrained to move between two points on the x-axis, where it can only take on certain discrete energy values due to the formation of standing waves. We will derive the relationship between the speed of the particle and its mass.
### Step-by-Step Solution:
1. **Understanding the Standing Wave Condition**:
The particle is confined between \( x = 0 \) and \( x = a \). The boundary conditions for standing waves require that there are nodes at both ends. The wavelength \( \lambda \) of the standing wave can be expressed in terms of the quantum number \( n \):
\[
a = \frac{n \lambda}{2}
...
When a particle is restricted to move along x-axis between x=0 and x=a , where alpha if of nenometer 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)/(2m) . 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. Use the model described above to answer the following three questions for a particle moving along the line from x=0 to x=alpha . Take h=6.6xx10^(-34)Js and e=1.6xx10^(-19) C. Q. 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=a , where alpha if of nenometer 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)/(2m) . 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. Use the model described above to answer the following three questions for a particle moving along the line from x=0 to x=alpha . Take h=6.6xx10^(-34)Js and e=1.6xx10^(-19) C. Q. If the mass of the particle is m=1.0xx10^(-30) kg and alpha=6.6nm , the energy of the particle in its ground state is closest to
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CENGAGE PHYSICS ENGLISH-PHOTOELECTRIC EFFECT-Linked Comprehension Type
