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Wave Optics

Question
CBSEENPH12039148
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Using Bohr’s postulates, obtain the expression for the total energy of the electron in the stationary states of the hydrogen atom. Hence draw the energy level diagram showing how the line spectra corresponding to Balmer series occur due to transition between energy levels.

Solution

According to Bohr’s postulates, in a hydrogen atom, a single electron revolves around a nucleus of charge +e. For an electron moving with a uniform speed in a circular orbit on a given radius, the centripetal force is provided by the Coulomb force of attraction between the electron and the nucleus.

Therefore, 
fraction numerator m v squared over denominator r end fraction space equals space fraction numerator 1 space left parenthesis Z e right parenthesis space left parenthesis e right parenthesis over denominator 4 pi epsilon subscript o r squared end fraction                        ... (1)
rightwards double arrow m v squared space equals fraction numerator 1 space over denominator 4 pi epsilon subscript o end fraction fraction numerator Z e squared over denominator r end fraction So comma space Kinetic space energy comma space straight K. straight E. space equals space 1 half m v squared K. E equals fraction numerator italic 1 over denominator italic 4 pi epsilon subscript o end fraction fraction numerator Z e to the power of italic 2 over denominator r end fraction 
Potential energy is given by, P.E = fraction numerator 1 over denominator 4 pi epsilon subscript o end fraction fraction numerator left parenthesis Z e right parenthesis space left parenthesis negative e right parenthesis over denominator r end fraction space equals space fraction numerator 1 over denominator 4 pi epsilon subscript o end fraction fraction numerator left parenthesis Z e right parenthesis space left parenthesis negative e right parenthesis over denominator r end fraction
Therefore, total energy is given by, E = K.E + P.E = Error converting from MathML to accessible text.
E = negative fraction numerator 1 space over denominator 4 pi epsilon subscript o end fraction fraction numerator Z e squared over denominator 2 r end fraction comma space is space the space total space energy.

For nth orbit, E can be written as En, 
straight E subscript straight n space equals negative space fraction numerator 1 over denominator 4 πε subscript straight o end fraction fraction numerator Ze squared over denominator 2 straight r subscript straight n end fraction                           ... (2)
Now, using Bohr's postulate for quantization of angular momentum, we have
mvr space equals space fraction numerator nh over denominator 2 straight pi end fraction
rightwards double arrow straight v space equals space fraction numerator nh over denominator 2 πmr end fraction

Putting this value of v in equation (1), we get
Error converting from MathML to accessible text. 

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Some More Questions From Wave Optics Chapter

Answer the following questions, which help you understand the difference between Thomson's model and Rutherford's model better.
Is the probability of backward scattering (i.e., scattering of α-particles at angles greater than 90°) predicted by Thomson's model much less, about the same, or much greater than that predicted by Rutherford's model?

Answer the following questions, which help you understand the difference between Thomson's model and Rutherford's model better.
Keeping other factors fixed, it is found experimentally that for small thickness t, the number of α-particles scattered at moderate angles is proportional to t. What clue does this linear dependence on t provide?

Answer the following questions, which help you understand the difference between Thomson's model and Rutherford's model better.
In which model is it completely wrong to ignore multiple scattering for the calculation of average angle of scattering of α-particles by a thin foil?

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