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MORE QUANTUM MECHANICS, HEISENBERG’S UNCERTAINTY PRINCIPLE

Let us return to the 1-dimensional quantum well problem and calculate the average momentum according to equation 1.12. Application of this equation leads to (in 1-dimension)

since the integrand is an odd function ofx. This result should not surprise us. In section 1.3 we found that the momentum is ftk with к = nnIL and with n taking both positive and negative integer values, so for each value of Tik there is always its negative as a permitted value of the momentum . So if repeated measurements of momentum are made they can always be grouped as pairs of numbers of opposite sign and the average will give p= zero.

This is an example of the strange two-fold rule of Quantum Mechanics. Prior to measurement of momentum of an electron with energy Тг2к2/2m the electron behaves like a string, a wave, and one can only say that the momentum of the electron is either hk or -Tik. Upon measurement, it will only come out as one of them (but never zero) and exhibit particle behaviour. Prior to measurement we can only talk about a range of possible values, only after measurement do we get a definite value. We saw this principle in connection with the two-slit experiment and the measurement of position, but it holds for all physical variables. Prior to measurement the electron could be anywhere according to a probability given by the modulus of its wavefunction squared. After measuring it on a screen it has a definite position and behaves like a particle. The same principle holds for momentum and any physical variable in any system.

It is worthwhile discussing further the comparison of the quantum well to the free particle case. We saw in the first section of this chapter that a free electron in vacuum has a wavefunction of the form e'kr. This wavefunction corresponds to a definite energy E = ti2k2/2m and a definite momentum Tik. One may wonder then why in the case of the quantum well with V(x) = 0 for 0 ^ x ^ L we have two values of momentum hk and -hk. The answer is that V(x) is not zero everywhere but only in the well. Outside the well the potential V is infinite which forces the electrons to bounce back at the well walls producing two running waves in opposite directions which in turn produce one standing wave as can be seen mathematically from the identity lisinkx = e,kx - e~,kx.

We now observe that the values hk are the eigenvalues of the momentum operator in one dimension. Indeed

with e'kx being the eigenfunctions. This can be extended to 3-dimensions

We now generalize this observation and state that the possible results of measurement of any physical quantity are the eigenvalues of an operator 0(p,r) corresponding to this physical quantity. This operator is constructed by substituting for the classical momentum p the operator (—ihV) and leaving the position vector r as it is, i.e. by the same rules as for the hamiltonian. Furthermore, 0(p,r) is hermitian as is the hamiltonian. We will not need any other operators apart from the ones we have already introduced. However, it is helpful to know that this is a general property of quantum mechanics and it does not pertain to momentum only, although the latter forms the main part of the current density which is our main quantity of interest. These issues are further developed in Appendix A. The content of this Appendix is not necessary for the understanding of the rest of this book.

The above discussion of momentum brings us to the famous Heisenberg’s uncertainty relation. We saw that in a 1-dimensional quantum well of size L, the most probable position

of an electron is at the center, x = — so that the uncertainty in its position is Ax = LI2. If

we make a momentum measuremetnt at energy E„ (i.e. of a stationary state), we can get tik„ or -hk„, i.e. the average is zero and the deviation from the average = the uncertainty Ap„ = Тшп IL. If we multiply these two quantities together we get

Since the minimum value of the integer n is 1 we get

This can be generalized for any physical system (not only for the 3-dimensional quantum well) to the following three relations

The above relationships are not intuitive generalizations from the 3 dimensional rectangular quantum well but can be proved from the mathematical properties of the eigenvalues of linear hermitian operators. As the name suggests, it was Heisenberg who achieved that. Now, in analogy with the above relationships we also have the corresponding uncertainty relation for energy and time. We say “the corresponding” because the product of energy and time has the same physical dimension as the product of momentum and length, i.e. the dimension of action, that of the physical constant ft. Therefore, we have

where At is the uncertainty of time (duration of measurement) and AE is the uncertainty in energy E. We will not occupy ourselves with this matter anymore as these relations will not be of direct use to us later.

A last point we would like to make is that in the free electron case the hamiltonian p212m and the momentump commute as operators and hence they have common eigenfunctions, the e'kx. When this happens in quantum mechanics for any two observables, the corresponding classical variables—i.e. the eigenvalues—can be measured simultaneously without error. This is the case of an electron in vacuum where both energy and momentum are known exactly.

 
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