Question

Question #4: (25 points total) In this problem, you are going to walk you through a brief history of quantum mechanics apply the principles of quantum mechanics to a physical system (free electron) 1900 Planck's quantization of light: light with frequency v is emitted in multiples of E hv where h 6.63x10-341.s (Planck's constant), and h =hw h 1905 Einstein postulated that the quantization of light corresponded to particles, now called photons. This was the correct way to analyze the photoelectric effect -the emission of electrons when light hits a surface. In his work on relativity, he showed that a photon with momentum p has an energy E pc, where c 3.00x10m/s. Therefore, p hk 1913-Bohr: electrons in atoms have wavelike properties, which explains the quantized energy levels of hydrogen atoms 1924 De Broglie: all particles have wavelike properties (wave-particle duality), so that E hv ha and p hk 1926 Schrödinger's equation - U(x)x)-E (x) where v(x) is the wave function for a particle with energy E, moving in the presence of a conservative force, with corresponding potential energy Ux) 2m dx 1927- Heisenberg's uncertainty prineiple: it is impossible to make precise determinations of a coordinate or a momentum of a particle simultaneously AxAp 2 h where Ax is the uncertainty on x and Ap,is the uncertainty on the momentum p Let's now consider a free electron (U-0). By analogy to the traveling waves we studied before, we can write its wave function as x)sin (kx -wt) Part A (10 points): Show that this wave function satisfies Schrödinger's equation if E energy corresponds to its kinetic energy) (its total 2m

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Answer 1

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