3. Galilean invariance of the free Schrodinger equation. (15 points) Show that the free-particle one-dimensional Schrödi

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3 Galilean Invariance Of The Free Schrodinger Equation 15 Points Show That The Free Particle One Dimensional Schrodi 1 (107.28 KiB) Viewed 44 times

3. Galilean invariance of the free Schrodinger equation. (15 points) Show that the free-particle one-dimensional Schrödinger equation for the wavefunc- tion V (x, t): at h2 32 V ih- at is invariant under Galilean transformations 2m ar2 X' = x – ut, t' = t. = By this we mean that there is a \'(x', t') of the form L' (x”, t') f(x, t) V(x, t), where the function f(x, t) involves x, t, ħ, m and v, and such that I' satisfies the corresponding Schrödinger equation in primed variables. ay' ih ət' hp 02 T 2m ax/2 (a) Find the function f(x, t). (Hint: Note that the function f(x, t) cannot depend on any observable of V; it is a universal function that is used to transform any V. Thus if V is a (single) plane wave, f cannot depend on its momentum or its energy.) (b) Demonstrate that the plane wave solution (x, t) A e{(kx-wt) transforms as expected. In other words, give li' and show that it represents, in the primed reference frame, a particle with the expected momentum and energy.