A. Suppose a particle starts out (t = 0) in the ground state of the initial well:
\ll(x, 0) = l sin (~x).
Show that the expansion coefficients can be written in the form
21rr . , Cn = - e-,c, z- sin(nz) sin(z) dz,
l[ 0
[10.5]
where a = mva/ 2n2n is a dimensionless measure of the speed with which
the well expands. (Unfortunately, this integral cannot be evaluated in terms
of elementary functions.)

B. Suppose we allow the well to expand to twice its original width, so the
"external" time is given by w(Te) = 2a. The "internal" time is the period of
the time-dependent exponential factor in the (initial) ground state. Determine
Te and T;, and show that the adiabatic regime corresponds to a « 1, so that
exp(-iaz2) ~ 1 over the domain of integration. Use this to determine the
expansion coefficients, en. Construct \ll(x , t), and confirm that it is consistent
with the adiabatic theorem

  sequence  is a continuous and distinctive band of  stars  that appears on plots of stellar  color  versus  brightness. these color-magnitude plots are known as  hertzsprung–russell diagramsafter their co-developers,  ejnar hertzsprungand  henry norris russell. stars on this band are known as  main-sequence stars  or  dwarf stars. these are the most numerous true stars in the universe, and include the earth's  sun.after condensation and ignition of a star, it generates  thermal energy  in its dense core region through  nuclear fusion  of  hydrogen  into  helium. during this stage of the star's lifetime, it is located on the main sequence at a position determined primarily by its mass, but also based upon its chemical composition and age. the cores of main-sequence stars are in  hydrostatic equilibrium, where outward thermal pressure from the hot core is balanced by the inward pressure of  gravitational collapse  from the overlying layers. the strong dependence of the rate of energy generation on temperature and pressure to sustain this balance. energy generated at the core makes its way to the surface and is radiated away at the  photosphere. the energy is carried by either  radiation  or  convection, with the latter occurring in regions with steeper temperature gradients, higher opacity or both.the main sequence is sometimes divided into upper and lower parts, based on the dominant process that a star uses to generate energy. stars below about 1.5 times the  mass of the sun  (1.5  m☉) primarily fuse hydrogen atoms together in a series of stages to form helium, a sequence called the  proton–proton chain. above this mass, in the upper main sequence, the nuclear fusion process mainly uses atoms of  carbon,  nitrogen  and  oxygen  as intermediaries in the  cno cycle  that produces helium from hydrogen atoms. main-sequence stars with more than two solar masses undergo convection in their core regions, which acts to stir up the newly created helium and maintain the proportion of fuel needed for fusion to occur. below this mass, stars have cores that are entirely radiative with convective zones near the surface. with decreasing stellar mass, the proportion of the star forming a convective envelope steadily increases. main-sequence stars below 0.4  m☉undergo convection throughout their mass. when core convection does not occur, a helium-rich core develops surrounded by an outer layer of hydrogen.in general, the more massive a star is, the shorter its lifespan on the main sequence. after the hydrogen fuel at the core has been consumed, the star  evolves  away from the main sequence on the hr diagram, into a  supergiant,  red giant, or directly to a  white dwarf.

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