The time-dependent wave function "(x, t) for
a free particle of mass m is given by:
30
(##)
"(x,t)=ei p x! Et =eip x.e!iEt (*)
where p and E are real constants.
The Potential Energy of a free particle is zero everywhere, in
other words, V(x) = 0. Thus the
wavefunction "(x,t) given above is a solution of the
time-dependent Schrödinger equation for a potential
energy V(x) = 0, given by the expression:
- The Time Dependent Wave Function X T For A Free Particle Of Mass M Is Given By 30 X T Ei P X Et Eip X E I 1 (84.48 KiB) Viewed 59 times
3) The time-dependent wave function P(x,t) for a free particle of mass m is given by: ¥(x,1)= ei(#x-#) = ctx.e-it where p and E are real constants. The Potential Energy of a free particle is zero everywhere, in other words, V(x) = 0. Thus the wavefunction P(x,t) given above is a solution of the time-dependent Schrödinger equation for a potential energy V(x) = 0, given by the expression: in-Y (x,t) "(x,1) = n² ² Y(x,1) (**) ді 2т дх2 Using the expression for P(x,1) given in equation (*), calculate Y*Y. b. Using the expression for P(x,t) given in equation (*), find i Yin terms of a real ді constant times V. 22 Using the expression for 4(x, t) given in equation (*), find -Y in terms of a real dx2 constant times Y. d. Plugging in your results in (6) and (c) into the free particle Schrödinger equation p? (**) above, show that the equation yields E = where E and p are the 2m' constants in the exponent of P(x,t). a. c. (Notice that, curiously, this is the classical expression for the kinetic energy of a free particle with linear momentum p = mv)