# Answers for $$\S5$$ QUIZ From Problem Sets selected for BSARCIST Workshop

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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?

(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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