Substitute the quantized radius in mvr=nℏ to get vn=e2/(4πϵ0ℏn). Coulomb potential energy is −e2/(4πϵ0rn) and the stipulated virial relation makes the total one half of this value. HenceThe negative sign means a bound state relative to a separated stationary proton and electron.
vn=nαc,En=−2(4πϵ0)2ℏ2n2me4=−n2Ry.
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