Dual Nature of Matter

JEE Main level, single correct, medium

81 of 200

Question 81 of 200

The de-Broglie wavelength associated with the electron in the n=4n=4 level is:

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Answer: (B) Four times the de-Broglie wavelength of the electron in the ground state

De-Broglie wavelength is given as

λ=hp\lambda =\dfrac{h}{p}

∵p22m=k⇒p=2mk∵\dfrac{{p}^{2}}{2m}=k\Rightarrow p=\sqrt{2mk}

⇒λ=h2mk\Rightarrow \lambda =\dfrac{h}{\sqrt{2mk}}

At ground level energy =E0={E}_{0} so

At n=4n=4 level energy =E0n2=E016=\dfrac{{E}_{0}}{{n}^{2}}=\dfrac{{E}_{0}}{16}

∴∴ at ground level

λ0=h2mE0{\lambda }_{0}=\dfrac{h}{\sqrt{2m{E}_{0}}}

At n = 4 level

λ′=h2mE016=4h2mE0{\lambda }^{'}=\dfrac{h}{\sqrt{2m\dfrac{{E}_{0}}{16}}}=\dfrac{4h}{\sqrt{2m{E}_{0}}}

⇒λ′=4λ0\Rightarrow {\lambda }^{'}=4{\lambda }_{0}

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