Chapter 4

Operators, Eigenstates, Eigenvalues and Schrodinger Equation

If Quantum Mechanics has not profoundly shocked you, you have not understood it yet.

—Niels Bohr

4.1  INTRODUCTION

We have learnt in the previous Chapter that it is consistent to visualize a microscopic particle (under all circumstances) as some form of (matter) wave. We shall represent these matter waves by the space- and time-dependent wave function ψ(r, t). In the previous chapter, we had represented the classical waves and wave packets by wave displacements y(r, t). In this chapter, we shall develop the concept of operators, operating on the wave function (or eigenstate) ψ(r, t) and providing information about their eigenvalues. We shall study how to write down the equation of motion which governs space and time development of the wave function ψ(r, t).

4.2  MEASUREMENT PROCESS AS OPERATOR OPERATING ON THE STATE FUNCTION/WAVE FUNCTION OF A PARTICLE HAVING DEFINITE LINEAR MOMENTUM

A microscopic particle may be manifested in the form of a infinitely extended plane propagating wave of a well-defined wavelength λ[so ∆λ = 0 or ∆p = 0 as p = h/λ gives ∆p = (h∆λ/λ²) (in magnitude) and ∆x = ∞, as the particle in the form of extended wave may be found any where in the region where the wave is present]. The particle may even be in the form of localized wave (packet) which, as we had seen in Chapter 3, is a superposition of a large number of plane waves having different wavelengths, thereby making ∆λ (or ∆p) non-zero and ∆x, the wave packet width finite (in fact, ∆x will be small if ∆λ is large). In each case, the Heisenberg’s uncertainty principle is being followed. In fact, the Heisenberg’s uncertainty principle has been obtained through the measurement process. Therefore, it seems logical to conclude: ‘any statement about a physical quantity of a system (i.e. the particle or the corresponding wave) has been made only after a measurement on the system has suggested so’. That further means every statement about the value of a physical quantity of the system is equivalent to ‘a measurement of that physical quantity of the system with the suitable experimental set up giving this particular value’.

For example, suppose we make a statement like this ‘a particle is moving in + x direction with its linear momentum having value p’. This simply means if one measures its linear momentum, it will have sharply defined value p [or sharply defined value λ(= h/p) as de Broglie relation dictates]. That further means we shall measure its linear momentum or wavelength using a two-slit arrangement which gives precise value of λ and gives no information about the position of the particle. So in quantum mechanics, making a statement about the value of the physical quantity of the system is equivalent to an experimental measurement process on the system for that physical quantity, as stated in the previous paragraph.

Let us consider the two-slit experiment with the beam of mono-energetic electrons (discussed in section 2.7). We can determine the wavelength λ of the electrons (in the beam) by measuring the fringe width β of the interference pattern on the screen, the distance d between the two slits and the distance D between the slit system and the screen.

A plane propagating classical wave of wave vector k and angular frequency ω propagating in + x direction is represented by wave displacement y(x, t) as

 

 

But for the moment, let us consider a plane propagating (matter/electron) wave represented by a complex function

 

Now let us see what the operator operating on the complex wave function ψ(x, t) gives

So the operator operating on the wave function ψ(x, t) gives us ħk multiplied by ψ(x, t).

Now we know that (using de Broglie relation) ħk is the linear momentum of the particle (electron in this case). So the operator operating on the wave function ψ(x, t), which is representing the moving electron in the beam, gives us information about the wave vector k of the wave or about the linear momentum p (= ħk) of the electron. To be more specific, if an electron wave is represented by the wave function

 

the operator operating on ψ(x, t) shall give

telling us that the linear momentum of the electron is 5ħ. On the other hand, when the mono-energetic electron beam represented by the wave function ψ(x, t) = Ae^{i(5x–ω₁t)} falls on two-slits, it shall produce interference pattern with a fringe width such that λ comes out to be 2π/5 from which we get k = 5 and linear momentum = 5ħ, the same as obtained by operation of the operator on wave function ψ(x, t).

So, on one hand, doing interference experiment with the electron beam described by the wave function ψ(x, t) = Ae^{i(5x–ω₁t)} (and finding the value of linear momentum of electron), and on the other hand, doing operator algebra with operator operating on the same wave function ψ(x, t) = Ae^{i(5x – ωt)} (and finding the value of the linear momentum of electron) seem to be equivalent. Therefore, so far as the information content is concerned, a particular operator operating on a particular wave function (by which the particle is described) is equivalent to a particular experiment done on the (matter) waves represented by the wave function. So, in a way, quantum mechanics may be called as operator mechanics, as mentioned in Chapter 1.

 

Table 4.1 A list of some physical quantities and corresponding operators.

Sl.No	Physical quantities	Corresponding operator
1.	x-component of linear momentum, px
2.	y component of linear momentum, py
3.	z component of linear momentum, pz
4.	Linear momentum, p	– iħ∇
5.	Kinetic energy,
6.	Energy of the particle, E
7.	Potential energy, V(r, t)	V(r,t)
8.	Total energy,

After studying the role of operation on the wave function ψ(x, t), let us see the role of one more operation . We find

Now ħω₁ is the energy of the particle which is represented by the plane wave of angular frequency ω₁. Therefore, the operator when operating on ψ(x, t) gives the value of energy of the particle.

So, what we have found in this section is that when we are dealing with free particles (i.e. particles propagating with definite, well-defined linear momentum) we may write their wave function as ψ(x, t) = A e^i(kx-wt). And the operators and operating on the wave function ψ(x, t) give the value of linear momentum and energy of the particles, respectively. We may summarize this whole in Table 4.1.

4.3  PHYSICAL INTERPRETATION OF WAVE FUNCTION ψ(r, t)

In Section 2.7, while interpreting the result of two-slit electron diffraction experiment, the number of electrons reaching a region around position y on the screen was compared with the intensity I of the resultant electron wave (i.e. with the square of the resultant amplitude of the wave) at that point y. Thevariation of number of electrons with y on the screen was found same as the variation of intensity I of the resultant electron wave. In Section 4.2, we have started denoting the electron wave propagating in x-direction by the wave function ψ(x, t). So, even without mentioning there, it was implicitly assumed in the analysis that the probability of finding an electron at a given location on the screen was proportional to the square of the modulus of the wave function of the electron wave |ψ(x, t)|². We may generalize it to the three-dimensional case, where the electron wave is represented by the wave function ψ(x, y, z, t). The probability of finding an electron at location (x, y, z) at time t is proportional to |ψ(x, y, z, t)|². In fact, this is similar to what was formulated as a fundamental postulate by Max Born in the form of statistical (or ensemble) interpretation of the wave function, which we describe as follows.

4.3.1  Statistical (or Ensemble) Interpretation of the Wave Function

According to Born’s postulate, the probability P(r, t) dr of finding the particle (in a measurement of the position of the particle) in a volume element dr (= dx dy dz) at point r = (x, y, z) is proportional to |ψ(r, t)|² dr (i.e. proportional to |ψ (x, y, z, t)|² dx dy dz). The quantity P(r, t) = |ψ(r, t)|² = ψ* (r) ψ (r) is known as position probability density. Whereas the wave function ψ(x, y, z, t), which may be a complex quantity, is not an observable (i.e. it can not be experimentally measured just as a physical quantity), the position probability density P(x, y, z) is an observable quantity and can be observed experimentally (but), in the following (special) way:

Let a particle be represented, in a given space and potential field, by the wave function ψ(x, y, z, t). Now imagine a large number of identical, separate single particle systems (say N) where each particle is in identical conditions, that is in identical volume (say V) under identical potential field. Now according to statistical (ensemble) interpretation of ψ, this ensemble of N (identical) systems is described by a single wave function ψ(x, y, z, t), and this wave function contains maximum information about these systems.

Let us be more specific. Let us divide the whole volume V (of each system) in a large number (say M) of equal smaller volumes (V/M). Let us call these small volumes as V₁, V₂, ... V_M and assume these as centered around positions r₁, r₂, ... r_M. Now measurements of the position of the particle are made separately on each of the N systems, say in the small region V_n around position r_n. In some of the systems, the particle will be found in region V_n, whereas in others, the particle is not found in V_n. Let L be the number of systems in which particle is found in region V_n. Now by the frequency definition of probability, the probability (P_n) that in a system, the particle is found in volume V_n (around position r_n) is L/M. Let us define the position probability density P(r, t) as the probability of finding a particle in the unit volume around position r. Then

According to Born’s postulate, this probability is governed by |ψ(r_n, t)|² in the way that

So Born’s postulate, effectively, connects |ψ(r_n, t)|² to the position probability density P(r_n, t) at r_n. In general

 

 

In Born’s statistical interpretation of the wave function, the wave function ψ(r, t) is associated with an ensemble of identical systems with |ψ(r, t)|² describing position probability density. However, we shall generally speak, (only) for convenience, of the wave function associated with a particular system.

Now, since the total probability of finding the particle somewhere must be unity, we must require

where integral extends over volume V in which the particle is confined. The wave function satisfying Eq. (4.10) is said to be normalized to unity. In fact, a class of wave functions for which the integral ∫|ψ(r, t)|² dr is finite are square integrable. A wave function, which is square integrable, can be multiplied by an appropriate (real or complex) constant to make it normalized (to unity).

4.4  SCHRODINGER EQUATION FOR A FREE PARTICLE

In Section 4.2 we had seen that the plane propagating matter wave described by the wave function

 

 

is representing a free particle propagating in positive x-direction with (precisely defined) linear momentum ħk. Now, if we apply Born’s statistical interpretation to the wave function of Eq. (4.2), we find the position probability density P(x, t) of the particle is given as

 

 

The probability density, that is, the probability of finding the particle in unit interval at position x, a constant quantity A², is independent of x. That simply means the particle is equally probable everywhere in a region where ψ(x, t) is non-zero. That also means the position of the particle [which is described by the wave function ψ(x, t) (Eq. 4.2)] is totally uncertain; ∆x = ∞, if the wave function is extended in an infinite region. This totally fits well with the uncertainty principle; the linear momentum of the particle is precisely known (to be ħ k) and the position is totally uncertain.

For a particle which is moving freely (i.e. no force working on the particle) in positive x-direction, the total energy is

This particle is described by a plane propagating matter wave of wave vector k = (p_x/ħ) and angular frequency ω = (E/ħ) [Eq. (4.2)],

 

Now operating the operator twice on ψ(x, t), we get

And operating the other operator on ψ gives

Hence, using the relation [Eq. (4.12)], we see that the wave function ψ(x, t) given by Eq. (4.13) satisfies the partial differential equation

This is called the time-dependent Schrodinger equation, which is written for a free particle moving along x-axis. We may generalize it for a three-dimensional case. The wave function of a free particle moving in a three-dimensional space in direction e_k (unit vector in the direction of wave vector k) may be written as

 

Generalizing the procedure used in obtaining Eq. (4.16), we may easily get

4.5  SCHRODINGER EQUATION FOR A FREE WAVE PACKET

We remember when we considered in Section 3.2 the superposition of a large number of (classical) plane waves, we found a wave packet having its amplitude non-zero only in a small region. We now imagine the formation of a wave packet by the linear superposition of large number of plane propagating matter waves each denoted by A e^i(kx–ωt). The resultant wave packet is denoted by the wave function ψ(x, t) given by [in analogy with Eq. (3.5)].

In terms of p and E (p = ħk, E = ħω), this equation may be rewritten as

Let us see what do we find if operator iħ (∂/∂t) operates on both sides of this equation.

[We have replaced here total energy E by (p² / 2m), as we are describing the motion of the particle in a potential-free region.]

 

So

(replacing p by ), and we get

Using Eq. (4.20) it gives,

This is the time-dependent Schrodinger equation for a wave packet represented by the wave function ψ(x, t) of a freely moving particle.

4.6  SCHRODINGER EQUATION FOR A PARTICLE IN A POTENTIAL

We have seen in previous sections that the operators and (which we denote by p_op and E_op respectively) when operating on free particle wave function ψ(x, t) give, respectively, the values of linear momentum and energy of the particle. The Schrodinger equation for a free particle.

may be written in the form

Now, the operator when operating on ψ(x, t) shall give value of kinetic energy of the free particle (p²/2m), which is really total energy of the free particle. For a particle in a potential V(x), the total energy would be p²/2m + V(x). Therefore, we can generalize Eq. (4.22) by replacing total energy of free particle by total energy of the particle in potential V(x), so

or

This is the generalization of the free particle Schrodinger equation (4.16). Equation (4.24) is the famous time-dependent Schrodinger equation for a particle in potential V(x), which was proposed by E. Schrodinger way back in 1926. It is the basic (one-dimensional) equation of motion of non-relativistic quantum mechanics. Its generalization to three-dimensional case gives

The operator inside the brackets on the R.H.S. of the Schrodinger equation (4.25) plays a very important role. It is called the Hamiltonian operator of the particle and is, generally, denoted by Ĥ. So the Schrodinger equation may be written as

with

4.7  EXPECTATION VALUE AND OPERATORS

In Section 4.2, we had discussed that the operator operating on a state function ψ(x, t) = Ae^{i(kx – ωt)} of a free particle gives us value of the linear momentum of the particle moving in x-direction according to the said wave function. We know that P(x, t) dx = |ψ(x, t)|² dx is the probability of finding the particle in the length dx about point x at time t, therefore, the quantity [–iħ (∂/∂x)] |ψ(x, t)|² dx, in a way, gives us information about the probable value of the linear momentum of the particle in the region x and x + dx at time t. And or gives us information about the probable value or average value or expectation value of the linear momentum of the particle in state ψ(x, t). Technically, we state it as ‘the value of x-component of the linear momentum of a particle in state ψ(x, t) is equal to the expectation value of the (corresponding) operator –iħ (∂/∂x) in state ψ(x, t)’, that is

Any physical quantity (e.g. position coordinates, components of linear momentum, kinetic energy, etc.) that can be measured experimentally is known as observable. What has been stated in Eq. (4.28) for the x-component of linear momentum may be stated for any observable. The value A of an observable in state ψ(r, t) is obtained as the expectation value of the corresponding operator Â, in state ψ(r, t), that is

The observables are represented by hermitian operators which have got real eigenvalues (see Chapter 7).

4.8  PROBABILITY CURRENT DENSITY: EQUATION OF CONTINUITY

In Section 4.3, it was concluded on the basis of probability interpretation of wave function that the wave function must be square integrable and, therefore, can be normalized (to unity), that is, satisfy the condition (4.10).

 

 

Here the integral extends over all space where the wave function of the particle ψ(x, t) is non-zero. Indeed once the wave function ψ(x, t) is normalized to unity at time t, it remains normalized (to unity) at all times. This can be said, as the square of the wave function |ψ(x, t)|² has been interpreted to be the position probability density. And this requires that the probability of finding the particle somewhere must remain unity at all times (if it is unity at t = 0). So the total probability is conserved at all times, that is

Let us consider the time-dependent Schrodinger equation (4.25) and its complex conjugate given by

Multiplying Eq. (4.25) from left by ψ* and Eq. (4.31) by ψ, we get, respectively

and

Taking the difference of these two equations, we have

or

or

where ρ = ψ* ψ is representing the position probability density and vector J is defined as

The relation [Eq. (4.33)] is in the familiar form of equation of continuity expressing, for example, charge conservation in electrodynamics or conservation of matter in hydrodynamics. We shall see that J(r, t) can be interpreted as probability current density. Equation (4.34a) may be written as

Here Re represents the real part of the quantity inside the bracket. Now, as summarized in Table 4.1, the quantity ħ▽/i represents the linear momentum operator and, therefore, ħ▽/im shall represent velocity operator . So Eq. (4.34b) may be written as

Now, in case of classical system of hydrodynamics, the particle number current density J is related to the particle number density ρ through the relation

In Eq. (4.34c), ψ*ψ (the position probability density) appears in place of particle number density ρ of Eq. (4.34d) and, therefore, J in Eq. (4.34c) should be called something like probability current density; it should give information about the number of particles crossing (perpendicular) unit area per unit time. It can be easily seen that the probability current density J(r, t) vanishes if the wave function ψ(r, t) is real (see Exercise 4.3) and J(r, t) is non-zero only for complex wave functions.

4.8.1  Probability Conservation and the Hermiticity of the Hamiltonian

The condition [Eq. (4.30)] really expresses the conservation (in time) of the normalization of the wave function. The condition [Eq. (4.30)] may also be obtained in terms of the Hamiltonian operator Ĥ. For this we shall use the time-dependent Schrodinger equation (4.26) and its complex conjugate

Let us start with L.H.S. of Eq. (4.30),

Using Eqs (4.26) and (4.35), it gives

So, we get

Operators which satisfy the condition [Eq. (4.36)] for all wave functions ψ of the wave function space [or linear vector space (see Chapter 7)] in which they act, are called Hermitian operators. It may be noted that we started with the Schrodinger equation (4.26) to arrive at Eq. (4.30) expressing the conservation in time of the normalization of wave function ψ(r, t). And again using Schrodinger equation (4.26) and its complex conjugate [Eq. (4.26)], the condition [Eq. (4.30)] was recast in the form of hermiticity condition [Eq. (4.36)]. Therefore, it is clear that the Hamiltonian [Eq. (4.27)] of a particle in a real potential V(r) is a Hermitian operator.

4.9  GAUSSIAN WAVE PACKET AND ITS SPREAD WITH TIME

Let us consider the wave packet given by Eq. (4.19) with the Gaussian form of the function g(k).

 

 

The wave function ψ(x, t), Eq. (4.19), becomes (taking A = 1).

At t = 0, we have

With k′ = k – k_o, we get

 

Using theory of complex variables, the integral may be evaluated to give

Here is the amplitude of the wave packet and e^ik_ox is its phase factor. The absolute square of ψ(x, 0), given by

has its pack value at x = 0. It may be easily noted that the wave packet is narrow for small values of a and broad for large values of a. Before we discuss propagation of this wave packet, let us see the relation of wave packet width with the width of Gaussian function g(k) used in the formation of the wave packet [Eq. (4.37)].

Below we show schematic plots of g(k) and B(x) for two different values of a. Figure 4.1 is for a = 4 and Figure 4.2 for a = 1. It is clear from Figure 4.1 that the (effective) width of the wave packet B(x) is quite large if only a narrow range of plane waves [i.e. narrow Gaussian g(k)] are taken in the formation of the wave packet. However, if a wide range of plane waves are used in the formation of wave packet, the resulting wave packet is narrow (i.e. sharply peaked) as in Figure 4.2(b).

Let us now study the time-development of the wave packet, that is, the propagation of the wave packet. The wave packet ψ(x, 0) has been formed of a continuous superposition of simple displacements e^ikx in a certain k - range with weight factor g(k) = e^{–a(k – k_o)2}. To see how this wave packet propagates in time, we should superpose, instead of displacements e^ikx, the propagating plane waves, e^{i[kx – ω(k)t]}, with the same weight factor g(k).

So,

Figure 4.1 Plots of g(k) and B(x) for a = 4

Figure 4.2 Plots of g(k) and B(x) for a = 1

It may be easily checked that if ω(k) is linear in k (just like in case of light waves propagating in vacuum, where ω = ck), Eq. (4.37) may be written as

 

This equation is similar to Eq. (4.38b), except that x in Eq. (4.38b) is replaced by (x – ct) here. Therefore, we can easily get

So, the wave function ψ(x, t) [= ψ(x – ct, 0)] is having the same shape as that of the starting wave-function ψ(x, 0); but, instead of being peaked at x = 0, it is now peaked at x = ct. Thus in case of light, the wave packet of light waves in vacuum propagates with the velocity of light c, without any sort of distortion.

However, the case of propagation of wave packet of matter waves may not be that simple. Let us reconsider Eq. (4.38a) representing propagation of the wave packet of particle waves. Let the wave packet, we consider, be strongly localized in k-space about k = k_o [i.e. let a be large in the Gaussian function e^{–a(k – k_o)} appearing in Eq. (4.38a): see Figs. (4.1) and (4.2)]. Then, it is not hard to see that the integral in Eq. (4.38a) will centre around k = k_o and it is reasonable to expand ω(k) about k = k_o; we are implicitly assuming ω(k) to be a slowly varying function of k. So we may write-

The first term being independent of k, is a constant. The quantity appearing in the second term is the group velocity [Eq. (3.12)] of the wave packet.

Let

and

then Eq. (4.38a) may be written as

where k – k_o = k′.

This equation looks like Eq. (4.38c) with a replaced by [a + i (βt) / 2] and x replaced by (x – v_gt). So just like Eq. (4.38c), Eq. (4.43) may be integrated to give

The absolute square of ψ(x, t) is

Obviously Eq. (4.44) represents a propagating wave packet with its peak moving with velocity v_g. It may, however, be noted here that the width of the wave packet is not fixed. Comparing the two equations (4.39c) and (4.44), one notices that the quantity a at t = 0 [in Eq. (4.39c)] is replaced by at time t [in Eq. (4.44)]. So the width of the wave packet grows larger as it propagates, that is, the wave packet spreads as it propagates.

4.10  WAVE FUNCTION IN MOMENTUM SPACE

We had seen in Section 4.5 that the wave function

is the most general solution of the Schrodinger equation

We shall see that the normalization constant B in Eq. (4.20) is . We would like to discuss the physical significance of function ϕ (p) [in Eq. (4.20)]. Let us firstly consider Eq. (4.20) at t = 0. We have

Using the inverse Fourier transform (Appendix B1), we have from Eq. (4.45a)

ϕ (p) is called the Fourier transform of function ψ(x). From Eq. (4.46), we may write

Now

So

The result [Eq. (14.47)] simply states that if a function ψ(x) is normalized to unity, its Fourier transform ϕ (p) is also normalized.

Let us now consider the expectation value of linear momentum operator :

or

This result suggests that |ϕ (p)|² dp may be interpreted as probability for finding the particle having momentum in between p and p + dp. That means |ϕ (p)|² is the probability density for finding the particle with momentum p. Therefore, ϕ (p) may be interpreted as the wave function in momentum space.

We may generalize the definition of ϕ (p) [Eq. (4.45a)] for a time-dependent position space wave function ψ(x, t) [which is the solution of the time-dependent Schrodinger equation (4.25) of a particle in presence of a potential field] as follows:

with its inverse Fourier transform

It may be easily checked that the relations similar to the relations [Eqs (4.47) and (4.49)] hold for time-dependent momentum space wave function ϕ (p, t), that is,

 

(see Exercise 4.4).

We see a sort of symmetry between functions in x – and p – space, if we re-look at some of the above equations, put below:

One may wonder, whereas p has operator representation – iħ (∂/∂x) [Eq. (4.55)] in x-space, whether x has the corresponding operator representation in p-space. In fact, it does. It may be easily seen (Exercise 4.5) that x has the operator representation

in momentum space. That means

4.11  THE EHRENFEST THEOREM

The Bohr’s correspondence principle (Appendix A3) suggests that whenever the distances and linear momenta involved in describing the motion of the particle are large enough to ignore the uncertainty principle, the motion of the wave packet (describing the particle) must coincide with the classical motion of the particle. In fact, a calculation due to Ehrenfest demonstrates that Newton’s laws of motion in the form

are satisfied by the average motion of a wave packet described by a wave function ψ (which is a solution of the Schrodinger equation). For simplicity, let us consider x-components of r and p. The expectation value of position x of a wave packet described by the wave function ψ(r, t) is given as

Differentiating Eq. (4.58), we get

Using Schrodinger equation (4.25) and its complex conjugate [Eq. (4.31)], Eq. (4.59) gives

The second contribution to the above integral may be written with the help of Green’s first identity⁽¹⁾, as

As the wave function vanishes at large distances, the first integral, which is over infinite bounding surface S, vanishes. So we have

 

 

Again using Green’s first identity, we have

The surface integral vanishes and Eqs (4.61) and (4.62) give

 

 

With this result, Eq. (4.60) gives

or

In a similar way, we may calculate the time rate of change of as

Using Eqs (4.25) and (4.31), it gives

Applying Green’s second identity⁽¹⁾ on the first integral and realizing that ψ and ∂ψ/∂x vanish at large distances, the first integral vanishes. So Eq. (4.66) gives

Equations (4.65) and (4.67) are really the quantum counterpart of the x-component of classical equations (4.57). More generally, we may write

and

The Eqs (4.68), known as Ehrenfest theorem, look like Newton’s law for expectation values and confirm to the correspondence principle.

4.12  THE UNCERTAINTY RELATIONS (REVISITED)

4.12.1  Uncertainty of Expectation Values

Let us consider a physical system [say a particle in a potential V(r)] in state ψ(r). We may be interested in finding out value of a physical quantity of the system (say, value of the kinetic energy of the particle) when it is in state ψ(r). For this we shall pick-up an operator corresponding to the physical quantity (e.g. for kinetic energy, the corresponding operator is . Therefore, following the arguments of Section 4.7, the ‘average’ or ‘expected’ result of a measurement of kinetic energy T of a particle in state ψ(r) is the expectation value of kinetic energy operator

The statistical (i.e. ensemble) interpretation of the wave function dictates the following: Let us prepare a large number (N) of identical systems (a particle in the present example) each in state ψ(r) and measure value of kinetic energy of each particle (separately). Let the values of kinetic energies be T₁, T₂,··· T_N. The average value of these quantities, that is shall be the same as that given by Eq. (4.69), that is

As mentioned above, when measuring the value of the physical quantity on the members of the ensemble [in the present example, it is kinetic energy of a particle in state ψ(r)], this value is not the same for all members of the ensemble. Therefore, there is a spread of values and the ensemble average is T. Let us now generalize it to any physical quantity (or observable) say A and the corresponding operator Â. The amount of spread (i.e. the uncertainty in the quantity) may be expressed in terms of the squares of the deviations:

 

 

or

 

In explicit integral form the uncertainty or spread or root mean square deviation is nothing but

Here the physical meaning of Eqs (4.71) and (4.72) is quite clear. Here quantity <Â> is the average of values of measurement of physical quantity A on a large number of identical systems. ∆A is a measure of the spread or uncertainty, defined through the root-mean-square (rms) deviations, in the values of the measurements.

4.12.2  Commuting and Non-commuting Operators

Let us consider two operators  and . Let us choose these operators from Table 4.1 as

and find out the commutator of  and [defined as . As discussed earlier, in quantum mechanics, operators operate on some state function of the system: only then we get some meaningful result. So to find out value of the commutator , let us operate these operators on the state function ψ(r).

so we get

or

Similarly, one may get

Let us take  and as

With a similar procedure

∴

Next consider  and as

We may easily check that

and

Similarly, it can be seen that the linear momentum operator commutes with the kinetic energy operator and with Ĥ, the Hamiltonian of the free particle. The operators like and Ĥ, which commute, are compatible. For a free particle, quantum mechanics allows linear momentum p and energy E to be simultaneously specified.

The value of the commutator of any two operators may be zero or non-zero. We may see from Eqs (4.75) and (4.77) that the commutator is non-zero for pair of operators [e.g. for (i) and for (ii) t and E] where the corresponding two quantities can not be measured simultaneously accurately. As we recollect, measurement of x-position of a particle adds uncertainty in its x-component of linear momentum (and vice versa), but does not add any uncertainty to y-component of its linear momentum. So uncertainty principal is there for two quantities (called as conjugate quantities) whose corresponding operators have non-zero commutator.

4.12.3  The Generalized Uncertainty relations

We know that the results of a physical measurement of some physical quantity of a system are real numbers. This means that the corresponding operator must have its expectation value to be real definitely. That also means that the eigenvalues of the corresponding operator (operating on some state function of the system) must be real definite. The Hermitian operators have got this property. An operator  is Hermitian operator if the expectation value is such that following condition is satisfied:

or in matrix notation

We shall discuss more about it in Chapter 7.

We noticed in the previous sub-section 4.12.2 that the uncertainty in the results of measurements in a pair of quantities is related with the non-commuting nature of the pair of corresponding operators. So if for two physical quantities (observables) A and B of a system, the corresponding operators  and do not commute, then the uncertainties in the values of the quantities are related with each other. In fact, in such a case, the spread in the values of one set of measurements of quantity A (associated with operator Â) is related with the spread in the values of quantity B (associated with non-commuting operator ). This is really the uncertainty relation, which we intend to discuss below by considering a pair of operators  and which do not commute.

Let us denote the Hermitian conjugate of operator  as Â^† (see Chapter 7) Then we can easily see that

so

 

 

Let us denote

then

 

 

It is to be noted that ∆A is the rms deviation.

Now

It may be noted here that the quantities like <Â> are numbers (and not operators).

Let us consider a linear combination of operators  and and denote it by

where λ is real. If both  and are Hermian then

so

or

It is clear that each of the three terms in the above equation has to be real and, therefore, the last term must be pure imaginary or zero. The minimum value of L.H.S. of Eq. (4.88) may be obtained by varying λ such that

which gives

or

With this value λ, Eq. (4.88) gives

or

In this equation, let us replace  and by δ and , we get

with Eq. (4.85) this gives

Using Eq. (4.84), we get

This may be called as the generalized uncertainty relation. We may consider an example of a particle moving in x-direction. The uncertainty in its position and linear momentum may be found using relation [Eq. (4.92)] by taking the operators  and as – iħ(∂/∂x) and , respectively. We know value of the commutator . So the uncertainty relation for momentum and position operators from Eq. (4.92) can be written as

or

Similarly uncertainty relation for energy and time operator becomes

4.13  THE (RESULTING) QUANTUM LOGIC

Based on the developments in quantum mechanics, discussed above, Birkhoff and von Neumann argued that our day-to-day logic, termed as ‘classical logic’, should be revised to deal with quantum situation. In fact, many other workers have also proposed simple revisions on classical logic in order to incorporate quantum reasoning. At the outset, a proposal of revising our day to day logic may appear counter to our common sense and universal beliefs. However, the enormous success of quantum mechanics in explaining almost all phenomenon of physical systems at quantitative level, compels one to doubt the basis of earlier ‘classical reasoning’.

Let us start with the statement

 

 

where alphabets, A, B, C denote (real or complex) numbers and symbols · and + denote respectively the ‘multiplication’ and ‘addition’ operations. With this dictionary, the statement [Eq. (4.95)] mentions simply the distributive law (of multiplication) of numbers. Next, let the symbols · and + denote the basic logical operations ‘and’ and ‘or’. Then statement (4.95) gives following logical relation.

 

 

We take now A, B, C for the following statements:

Putting statements (4.97) on L.H.S. and R.H.S. of (4.96), we get

 

The statement (4.98a) is logically equivalent to the statement (4.98b). This equivalence may be termed as ‘classical logic’.

But now let us look at the logical relation [Eq. (4.96)] with the following statements for A, B, C:

With A, B, C of (4.99), the L.H.S. and R.H.S. of (4.96) give

 

It will be discussed in Chapter 12 that just like the case of measurement of position and linear momentum of a particle, in the case of spin particle, it is not possible to measure simultaneously the two components (say S_x and S_y) of spin angular momentum precisely.

Now, whereas the statement (4.100a) is valid quantum mechanically, the statement (4.100b) is not only inequivalent to (4.100a) but also operationally invalid. In fact, there is no experimental procedure which can simultaneously determine x- and y-components of the electron spin. Therefore, the L.H.S. and R.H.S. of logical relation [Eq. (4.96)] are not equivalent for quantum mechanical case [Eq. (4.99)]. This failure of ‘classical logic’ for quantum systems suggests that the classical logic has to be revised to be applicable to quantum mechanical case. The revision of logic should be in giving new procedures to operate the logical connectives ‘and’ and ‘or’, the details of which are beyond the scope of this book.

EXERCISES

Exercise 4.1

Find the commutators

• 
• 
• 

Exercise 4.2

Show that

Exercise 4.3

Find probability current density in case a particle is in state

• ψ(x, t) = A sin k x e^{–iħk2t / 2m}
• ψ(x, t) = A e^{i (k x – ħk2t / 2m)}

Exercise 4.4

Prove relations (4.52) and (4.53) using the definition (4.51) of ϕ (p, t).

Exercise 4.5

Show that x has operator representation in momentum – space.

Exercise 4.6

A particle is described by the normalized wave function.

• Find the normalization constant A.
• Find < x > and < x² >
• Find the value of x where probability density P(x) = |ψ(x)|² is maximum
• Find the corresponding momentum space wave function ϕ(p)
• Using ϕ(p) calculate and  

Exercise 4.7

Find the momentum space wave function ϕ(p, t) for a particle which has position space wave function ψ(x, t) = A e^–ax2 e^{–i E_o t / ħ}.

SOLUTIONS

Solution 4.1

•  
  
•  
  
•  
  

Solution 4.2

Proceeding in this way, we shall get

Solution 4.3

The probability current density J (x, t) is given as

•  
  
•  
  

Solution 4.4

L.H.S. of Eq. (4.52)

 

 

Putting expression of ϕ(p, t) from Eq. (4.51), we get

as

so

Solution 4.5

We have to show that

Using Eq. (4.50), we may write

Solution 4.6

•  
  

  ∴

        A = 2 (b)^3/2
•  
  
  
•  
  P(x) = 4 b³ x² e^–2bx

   

  For maxima,

  

  which gives maxima at

   

  x = 1/b
•  
  
  
  
•  
  
  

Let 

Then

Solution 4.7

Using the result from Appendix C6, we get

REFERENCES

1. Merzbacher, E. 1999. Quantum Mechanics, 3rd edn. New York: John Wiley.
2. Schiff, L.1968. Quantum Mechanics, 3rd edn. New York: McGraw-Hill.
3. Powell, J.L. and Crasemann, B. 1961. Quantum Mechanics. Reading, MA: Addison-Wesley.
4. Feynman, R.P., Leighton, R.B. and Sands, M. 1965. The Feynman Lectures on Physics, Vol. III. Reading, MA: Addison-Wesley.
5. Levi, A.F.J. 2003. Applied Quantum Mechanics. Cambridge: Cambridge University Press.
6. Gasiorowicz, S. 1996. Quantum Physics, 2nd edn. New York: John Wiley.
7. Ballentine, L.E. 1970. ‘The Statistical Interpretation of Quantum Mechanics’, Reviews of Modern Physics, 42: 358.
8. Whitaker, M.A.B. and Ishwar Singh. 1982. ‘Interpretation of Quantum Mechanics and Some Claimed Resolutions of the E P R Paradox’, J. Phys. A; Math. Gen., 15: 2377.
9. Deepak Kumar and Ishwar Singh. 1987. ‘Reality in Quantum Mechanics’, Physics Education, 3: 34.
10. de Espagnat, B. 1976. Conceptual Foundations of Quantum Mechanics, 2nd edn. Reading, MA: Benjamin.