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Answers for \(\S5 \) QUIZ From Problem Sets selected for BSARCIST Workshop

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Submitted by kapoor on Sun, 10/31/2021 - 17:49

When is a particle a free particle? How do you recognize a free particle, if a
statement about the wave function representing the  state of the particle is
given to you?

Question:

Several statements about the wave function of a particle are given below.  For each case, find out if the particle  must be a free particle or not?


Write your answer as
(a) TRUE,  if the particle must be free, and
(b) FALSE, if it need not be free
 

The wave function of the particle

ToHide: 

even
[Q1] satisfies the energy eigenvalue equation \[-\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} = E \psi(x); \]

[A.1]  TRUE ;

REASON

because the Hamiltonian in the energy eigenvalue equation is written as free particle Hamiltonian.

[Q.2] is Gaussian wave packet: \[ \psi(x)= N \exp(-\alpha^2 x^2);\]

[A.2] FALSE 

REASON


It  is a  wave packet that could constructed from eigen functions of any hermitian operator.
So it does not imply that the particle is free.

[Q.3] is a linear combination of sine and cosine wave function \[ A \sin k x + B \cos kx;\]

[A.3] FALSE

REASON

Just giving a wave function does not tell us if the particle if free or not.
One must have a way of specifying time dependence or the Hamiltonian.

[Q.4] is a ``plane wave'' \[ \exp(\pm{i} kx);\]

[A.4] FALSE

REASON

  • Wave function being plane wave does not guarantee that its  Hamiltonian of the system ( energy ) is free particle Hamiltonian.
  • It may be the wave function at some initial time. For example the wave function incident beam in a scattering experiment can be taken to be  free particle.
  • To decide if it is free particle or not, we must know how the wave function evolves with time.
[Q.5] satisfies Schr\"{o}dinger equation \[ i\hbar\frac{\partial \psi(x,t)}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\psi(x,t)}{\partial x^2}; \]

[A.5] TRUE

REASON

Because the wave function obeys Schrodinger equation;

The Hamiltonian is free particle Hamiltonian.

[Q.6] is normalizable: \[ \int_{-\infty}^\infty |\psi(x)|^2 \, dx < \infty. \]

[A.6]False

REASON

All vectors in a Hilbert space are square integrable. Hence this property alone has no information about the system.

Exclude node summary : 

n

REMARKS:

  • Whether a particle is free or not, this information is contained in the Hamiltonian.The constraint on the Hamiltonian should some how determine the Hamiltonian to be free.
  • So for example, giving wave function at a fixed time, does not tell me how it evolves in time, thereforethis information puts no restriction on the Hamiltonian.
  • This is like knowing velocity at a given time in classical mechanics does not have information about the forces.We have to know if velocity is changing with time, if yes, how it  is changing.

It will be a useful exercise to make statements will be  true, and which will be false for a free particle in quantum mechanics

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