A free particle is not subjected to any forces, its potential energy is constant.  Set V(r,t)=0, since the origin of the potential energy may be chosen arbitrarily.  The Schroedinger equation then becomes 

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Plane waves are possible solutions,

y(r,t)=Aexp(k×r-wt),

as long as  hw=h²k²/(2m).  

We have

,

using the de Broglie relations.

A plane wave represents a particle whose probability of presence is constant throughout space;

|y(r,t)|²=|A|²=constant.

But a plane wave is not square-integrable, it is not a proper solution.

The Schroedinger equation is a linear equation, the principle of superposition applies.  A linear combination of plane wave solutions is also a solution.  

will be a solution as long as for each k we have  hw_k=h²k²/(2m).  
Since k is a continuous variable, the most general solution is not a sum, but an integral;

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Such a wave function is called a three-dimensional wave packet and can represent any non-pathological square-integrable function.
(g(k) can be complex: g(k)=|g(k)|exp(ia(k)) ;  a(k) changes the phase of the plane wave.)
Proper wave functions of free particles are wave packets.

One-dimensional wave packets representing free particles

Let us first concentrate on one-dimensional systems.

is a superposition of plane waves traveling in the ±x direction.  At some instance of time, say t=0,

,

 ;

g(k) is the Fourier transform of y(x,0).

(Fourier series:  see appendix I, Cohen-Tannoudji)

Any piece-wise regular periodic function (finite # of discontinuities, finite # of extreme values) can be written as a series of imaginary exponentials.  Assume f(x) is a periodic function of x with fundamental period L;

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The coefficients C_m are given by

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In the limit  L ® ¥ this can be generalized to

,

where

;

f(x) and are Fourier transforms of each other.

Note: A Fourier transform is a linear transform;

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Consider a wave packet  

,

and assume |g(k)|is centered at k₀ where it reaches a maximum and has a width Dk.  Assume Dk<<k₀.  Assume g(k)=|g(k)|exp(ia(k)).  Such a wave packet represents a particle with fairly well defined momentum p (Dp<<p).

In the neighborhood of k=k₀ we may expand a(k) in a Taylor series expansion;

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We are only interested in a(k) in a small range of k about k₀.  Keeping only zeroth and first order terms we may write

,

and setting  

this becomes

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Here |g(k)| is nonzero only in a small interval about k₀.  Therefore we need to integrate only over this small interval of width ~Dk.  

In the integral |k-k₀| ~£ Dk . 

If we write exp(i(k-k₀)(x-x₀))=exp(i2p(k-k₀)/t) with  t=2p/(x-x₀) we find that for |t|<<Dk or |x-x₀|>>Dk^-1 the function exp(i(k-k₀₎(x-x₀)) oscillates very rapidly in the integration interval and the integral yields approximately zero.  

For x-x₀ @ 0  |t| ® ¥  and the function  exp(i2p(k-k₀)/t) @ 1 in the integration interval.  Then the integral yields its maximum value.  We therefore have

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is the position of the center of the wave packet.

As |x-x₀| increases |y(x,0)| decreases.  We expect the width of the wave packet Dx in coordinate space to be ~Dx=Dk^-1.  Then  DxDk ~³ 1.  This is the classical relation for two functions which are Fourier transforms of each other.  The product DxDk has a lower bound.  The exact value of this bound depends on the precise definition of Dx and Dk.

Assume we want to build a wave packet representing a particle which has a high probability of being observed at position r=0 in a small time interval Dt about t₀;

Here and y(0,t) are Fourier transforms of each other.  If y(0,t) is centered at t₀ with small width Dt, then  has width Dw  such that  DtDw ~³ 1.

The Heisenberg uncertainty relations:

Quantum Mechanics implies that it is impossible to know, at any given time, the position and the momentum of a particle to an arbitrary degree of accuracy.

Similarly:

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Heisenberg's Uncertainty Principle

Example:

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If the energy of the ground state is known exactly, then the energy of the excited state must be uncertain by DE.

We may write the Fourier transform of y(x,0) in the following way:

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Then

is the probability of finding the particle at a position between x and x+dx, and

is the probability of finding the particle with momentum between p and p+dp.  The constant C is the same in both expressions.  This follows from the Bessel-Parseval relation

(See appendix I, Cohen-Tannoudji).

For a plane wave y(x,0)=Aexp(ik₀x) we have Dk=Dp=0, g(k)=d(k-k₀), Dx=¥ .

Examples:

The momentum of the particle is on the order of p=mv=10^-18Js/m.  There are no practical restrictions on how accurately the particle’s momentum can be measured.

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But the uncertainty principle requires DpDx ³ h.  We therefore need h<<nh, which only holds if n>>1.  The uncertainty principle rejects the semi-classical Bohr model with Bohr orbits except for high Rydberg states.

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To estimate r₀ we let

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The quantitative agreement is accidental, only qualitative agreement should be expected.

Classically:

 

  E_min = -¥  at r₀=0.

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