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The de Broglie relations connecting the wave-particle dual nature as generalised from photons to all matter particles are given by $$E=\hbar\omega$$ and $$p=\hbar k$$ (Characterizing a single plane wave with wave parameters $\omega$ and $k$ .)

Are these relations true for a time dependent potential $V(x,t)$ And if yes then is there any proof, if not then can we say that de Broglie's relations are hypotheses?

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This relations are true for plane waves, i.e. for solutions of Schrödinger equation in absence of a potential. When a potential is present, they are generally not true (although there are some exceptions, such as Block wave solutions in periodic potentials).

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  • $\begingroup$ Suppose $V(x)$ is such that its maximum value is less than energy eigenvalue $E$. Then what happens to the de Broglie relations? $\endgroup$ Nov 23, 2020 at 8:54
  • $\begingroup$ Under some conditions one can use quasiclassical approximation and Bohr-Sommerfeld quantization rule, to reason in terms of De Broglie waves. But, as you mention yourself in the question - these relations are strictly true only for plane waves, which is not the case when one has potential. One needs a more complete theory in this case. $\endgroup$ Nov 23, 2020 at 9:16
  • $\begingroup$ But we can still have plane waves when we have constant potential $V_0$ $\endgroup$ Nov 25, 2020 at 20:56

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