To understand the principles discussed in chapter 2 and to use some of the mathematical results obtained in chapter 3 and this chapter, we will study the simplest possible system viz. the one dimensional free particle. The classical case of this problem is quite trivial as
CHAPTER 4. SOME SIMPLE EXAMPLES 31 it would give the solution to be a constant velocity trajectory. In quantum the problem is not as trivial and does merit discussion. It is to be noted that for a particle to show quantum behavior it must be small enough e.g. an electron.
The form of the SchrÄodinger equation tells us that the system is described completely by the hamiltonian H. From classical physics the form of the free particle hamiltonian is known to be
H = P2
2m; (4.20)
whereP is the momentum andmthe mass. In quantum,P is known to be an operator. We shall now try to predict the measurement of three common observables viz. momentum, energy and position.
4.2.1 Momentum
We already know P has continuous eigenvalues that can take values from minus to plus in¯nity. So if we start with some state jsi the result of aP measurement will be p with probability jhpjsij2 (postulate 4) if jpi is the eigenstate corresponding to the eigenvaluep.
As a result of the measurement the system will collapse into the state jpi. As an operator commutes with itself i.e. [P,P] = 0, it is easy to see that (equation 1.14)
[P; H] = 0: (4.21)
Hence, from theorem 4.2, P is a conserved quantity and subsequent measurement of mo-mentum on this system will continue to give the same valuep as long as the system is not disturbed in any other way. The state of the system staysjpi.
4.2.2 Energy
If jEi is an energy eigestate with eigenvalue E then the probability of measuring E in a state jsi would be jhEjsij2. As we are considering only conservative systems, energy is of course conserved and hence every subsequent measurement of energy will produce the same value E as long as the system is not otherwise disturbed. Now it can be seen that jpi is also an eigenstate of H (see problem 3).
Hjpi= P2
2mjpi= p2
2mjpi=Ejpi: (4.22)
Hence, the set of states jpi are the same as the set of states jEi. However, we choose to label these simultaneous eigenstates with p and not E. This is because, in E, they are degenerate. Two states with opposite momenta have the same value for E(=p2=2m).
CHAPTER 4. SOME SIMPLE EXAMPLES 32 In chapter 3 we saw that the position representation of jpi i.e. its wavefunction (for
¯xed time) is
ªp(x) =hxjpi=Aexp(ixp=¹h): (4.23) As this is also an eigenstate of energy, from equation 4.4 we see that the time dependence of this wavefunction is given by
ªp(x; t) =Aexp[i(xp¡Et)=¹h]: (4.24) This function is seen to be a wave with wavelength 2¼¹h=p and angular frequency E=¹h.
Historically, it was this wave form of the position representation of the energy eigenstates of a free particle that inspired the name wavefunction. In early interference type experi-ments this relationship between wave properties (wavelength and frequency) and particle properties (momentum and energy) was discovered.
Now one can see why the position representation has been historically preferred. Ex-periments like electron di®raction basically make position measurements on some given state. By making such measurements on several electrons (each a di®erent system) in the same state the probability distribution of position measurements is obtained. And this probability distribution is directly related to the position representation of a state as given by equation 3.4.
4.2.3 Position
The position operatorX is not a conserved quantity as it is seen not to commute with the hamiltonian. Using equation 3.7 and the properties of commutator brackets, one obtains
[X; H] = [X; P2
2m] =i¹hP
m: (4.25)
Hence, position measurements are meaningful only in certain types of experiments. A position measurement on a system can predict very little about subsequent position mea-surements on the same system even if it is not disturbed in any other way. However, in particle scattering type experiments, a position measurement is made only once on each particle. In such experiments position is measured for di®erent particles each in the same state to obtain information on the probability distribution of position measurements3. This is the kind of situation we will be interested in.
If a particle is in a state jsi, its position probability distribution isjhxjsij2. In partic-ular if jsi is a momentum (or energy) eigenstate, this distribution is jhxjpij2 which, from equation 3.20, is seen to be independent ofx. Hence, a particle with its momentum known
3For this, each particle needs to behave like a separate isolated system which means the density of particles must be low enough to ignore interactions amongst them.
CHAPTER 4. SOME SIMPLE EXAMPLES 33 exactly, is equally likely to be anywhere in space! This is a special case of the celebrated Heisenberg uncertainty principle (see appendix B).
To understand the unpredictable nature of nonconserved quantities, it is instructive to further analyze this speci¯c example of the position operator for a free particle. So we shall see what happens if repeated position measurements are made on the same free particle.
Whatever the initial state, the ¯rst measurement results in a value, say x1. This collapses the state to jx1i. Due to nonconservation of position, this state starts changing with time right after the measurement. If the ¯rst measurement is made at timet= 0, at later times t the state will bejx1it which can be found from the SchrÄodinger equation:
i¹h@
@tjx1it=Hjx1it: (4.26)
To observe the time development ofjx1i, as given by the above equation, it is convenient to expand it in energy eigenstates4 which in this case are the momentum eigenstates jpi. So, fort= 0 we write
jx1i = Z
hpjx1ijpidp: (4.27)
The time development of this state is given by (problem 2) jx1it=
Z
hpjx1ijpitdp; (4.28)
wherejpitis the time development of the energy eigenstate as given by equation 4.4. Then using equation 3.25 we get
jx1it= (2¼¹h)¡1=2 Z
exp(¡ipx1=¹h) exp(¡iEt=¹h)jpidp; (4.29) where E =p2=2m. Hence, at time tthe probability of measuring a valuex for position is given by jhxjx1itj2 where
hxjx1it= (2¼¹h)¡1=2 Z
exp(¡ipx1=¹h) exp(¡iEt=¹h)hxjpidp: (4.30) Using equation 3.20 andE =p2=2mthis gives
hxjx1it= (2¼¹h)¡1
This integral has meaning only in a limiting sense. If u(x1¡x; t) =
4Expanding in known eigenstates of a conserved quantity is convenient because its time dependence is simple. In particular, the time dependence of the energy eigenstates is already known
CHAPTER 4. SOME SIMPLE EXAMPLES 34 with a >0, then
hxjx1it= (2¼¹h)¡1lim
a!0u(x1¡x; t): (4.33) Computing the integral in equation 4.32 gives
u(x1¡x; t) = whereb= 2m¹ha. Hence, the probability of ¯nding the particle atx after timetis
jhxjx1itj2 = lim
To understand this physically we ¯rst consider a nonzero value forb. In that case we notice that the probability decreases with time if
jx1¡xj<
and increases at points farther out fromx1. We shall call the right hand side in the above inequality the inversion point. With time, the inversion point moves outwards from x1. This is sometimes interpretted as probability \°owing" outwards from the initial pointx1 somewhat like in di®usion.
If b = 0, equation 4.35 gives unusual results. At t = 0 the probability is still zero everywhere other thanx=x1. But even an in¯nitesimal time later, the probabilty becomes a nonzero value constant over all space and decreases with time as 1=t! This happens because with b = 0 the initial delta function wavefunction has in¯nite momentum (and hence, in¯nite velocity) components in ¯nite amounts. Therefore, parts of the probability can go to in¯nity instantaneously and then get lost giving a decreasing overall probability.
In the light of special relativity in¯nite velocity is not possible. This issue can be resolved only by introducing a relativistic quantum mechanics as will be done later.