Chapter 19

Theory of Measurement in Quantum Mechanics

To be without some of the things you want is an indispensable part of happiness.

—Bertrand Russell

19.1  INTRODUCTION

In earlier Chapters, we have gone through the development of quantum mechanics and through the applications of Schrodinger equation to a number of physical systems. The information about microscopic system, obtained by solving Schrodinger equation, is generally in the form of probability and can be verified experimentally only in a statistical way. It simply means we shall have to do identical experiments on a large number of identically prepared systems to verify the statistical results of Schrodinger equation. This is the famous statistical (or ensemble) interpretation of wave function.

Let us re-emphasize here that there is a fundamental difference between the statistical description of classical systems and that of quantum systems. To illustrate this point we consider the following example.

We first consider a classical particle in a one-dimensional double-well potential (as shown in Figure 19.1), for example, a small ball in a gravitational double-well. Suppose the ball is put by some one at the bottom of one of the wells; we do not know in which well the ball is. The information about the ball is only in terms of probability; the probability of the ball to be in either of the wells is . Suppose at time t = 0, we look at the wells and find the ball in well no. 2. We obviously conclude that the ball was there in well no. 2 at time t < 0 as well.

Figure 19.1 A classical particle in one-dimensional double potential well (with minimum energy)

Figure 19.2 A quantum particle in a one-dimensional double potential well [ground state ψ₀(x)]

Now consider a quantum particle in a similar one-dimensional double-well potential (Figure 19.2). To know the state of the particle, we shall solve time-independent Schrodinger equation. From our experience with one-dimensional problems, we can say that the particle in ground state is described by a wave function, say ψ₀(x), as shown in the figure. The wave function ψ₀(x) has larger amplitudes near the centres of the two wells. It decreases exponentially in the barrier regions separating the two wells and at the outer boundaries of the wells. The wave function mod square |ψ₀(x)|² gives the information of the position probability density P₀(x) of the particle. From this, we may find the probability of the particle to be found in the interval [x₁, x₂] as P₀(x)dx.

We may easily conclude that the probability of the particle to be found somewhere in well no. 1 is ~ and same for well no. 2. Now we do the following experiment; we take two microscopes, one probing the region of well no. 1 and the other of well no. 2. Two observers are (separately) observing through the microscopes. Say, observer no. 2 finds the particle in well no. 2 (and, obviously, observer 1 finds nothing in well no.1). What should be concluded from the observation that the particle is there is well no. 2. Can we conclude (just like that in classical case) that particle was there in well 2 even before the observer observed it? Clearly no; we cannot conclude this way. We know, before the observation, the particle was described by the wave function ψ₀(x), and, therefore, was described through the position probability density P₀(x). It is only the observation process (the interaction of microscope probe with the particle) that changed the wave function ψ₀(x) to a localized wave function (say delta-function) in well no. 2. This is the famous phenomenon of wave function collapse during the process of observation in quantum mechanics.

To conclude about the classical and quantum cases discussed above; In classical case, if the ball is found in well no. 2 upon observation, we conclude the ball was there in well 2 throughout the time. In quantum case, if particle is found in well no. 2 upon observation that does not mean the particle was there in well 2 throughout the time (before measurement); the particle was, in fact, spread throughout both wells (before the observation) in accordance with ψ₀(x).

It will be useful at this stage to reconsider the case of a particle in a one-dimensional potential box of width L with periodic boundary conditions (discussed in Section 5.5). The normalized wave function of the particle is

which gives the position probability density P_n(x) as

which is uniform. If we take the particle as an electron, its charge density ρ(x) [= (–e)P_n(x)] is also uniformly spread.

Now to check if the electron (and hence the electronic change) is really spread uniformly throughout the box length, we shall have to do measurements. For this, let us divide the width L in, say, 20 equal parts. Observations are made through microscopes in each of the 20 parts. Suppose the electron is detected in the 15th part (and obviously found absent in other parts). We cannot conclude from this observation that electron was present there (in the 15th part) even before it was observed. In fact, before the observation the electron was spread throughout the width L of the box, which can be verified in a statistical way. For example, if we have 100 identical one-dimensional boxes, each having an electron in its ground state and all 100 boxes are observed independently in the same way as mentioned above (dividing the width in 20 parts), we shall get following result.

Let N₁ be the no. of boxes, in which the electron is found in 1st part, N₂ in 2nd part, and so on. We will find that all these numbers are almost equal (to 5), confirming the constancy of the probability density.

19.2  PROCESS OF MEASUREMENT: A SIMPLE TREATMENT

It is a well-known fact that in any measurement process whether on a macroscopic system or on a microscopic system, the final observation of the quantity measured is always made at the macroscopic scale which is readable or observable by our naked eyes. The final observation could be either through a pointer or through a bright spot on the screen or through digits displayed on the meter etc. but it has to be at macroscopic scale. And so obviously in every type of observation, we use at least one classically describable stage. Therefore, it is clear that the quantum theory of the measurement process is nothing but a description of the relation between the state of the (quantum) system under investigation and the state of the classically describable part of the apparatus.

Let us proceed to study the process of measurement of the state of a quantum system S by an apparatus A. Before the measurement process starts, there is no interaction between system S and apparatus A. Therefore, the Hamiltonian operator before the measurement may be written as

 

 

where Ĥ_S(x) is the Hamiltonian of the system alone (and is a function of system variables x) and H_A(y) is the Hamiltonian of the apparatus alone (and is a function of the apparatus variables y).

Now, we denote the state of the apparatus by the wave function f(y, t). As the apparatus has to function in a classically describable way, the wave function f(y, t) should be in the form of a wave packet (but its description is much less precise than dictated by the uncertainty principle). As an example, y may represent the position of a voltmeter needle. The wave function of system S may be expanded in terms of the orthonormal basis states ϕ_m(x), as

with (unknown) coefficients a_m. Now, when the apparatus A starts interacting with system S, the total Hamiltonian shall include the interaction term and becomes

 

 

It is the interaction term Ĥ_I(x, y) which makes the measurement possible by introducing the correlation between the state of system S and that of apparatus A. Now the procedure of measurement should be such that the interaction is non-zero only for a certain time, so that the state of apparatus f(y, t) changes according to the state of the system S. After that the interaction becomes zero and the apparatus remains in the changed state for all time. Then any observer may consult the record of the apparatus without affecting the system S.

What we have mentioned above is a sort of impulsive measurement, where the interaction Ĥ_I is non-zero only for a short time but is strong. So changes in the total wave function

 

 

during that small time are only due to the interaction Ĥ_I(x, y). Thus during this time, the Schrodinger equation may be written as

Of course, before and after the interaction, the Schrodinger equation is

Let the observable under consideration be denoted by operator having eigenvalues m and the corresponding eigenstate ϕ_m(x). If the state of the apparatus A is to be correlated to the eigenstate of operator (of system S), Ĥ_I should depend on and y. Now, if Ĥ_I is in such a form that it is diagonal in the same representation in which is diagonal, that is if Ĥ_I and commute, then is a constant of motion. That simply means the matrix elements corresponding to the transition from one value of m to another will vanish. That further means, however strong H_I may be, there is no change in m though there may be large change in the complementary observable. So to get Ĥ_I diagonal when is diagonal, we should have,

This form of interaction Hamiltonian fulfils our objective of designing an apparatus with which the quantity m (eigenvalue of operator ) may be measured without itself suffering any change

Putting the form of Ĥ_I of Eq. (19.8) in Eq. (19.6), we get

The combined wave function ψ(x, y, t) which is governed by Eq. (19.9) represents the state of the system S and of apparatus A during the time of measurement. Therefore, we may have following expansion of wave function ψ(x, y, t):

where ϕ_m(x) are eigenfunctions of operator . Let us look at the information contained in the combined wave function ψ(x, y, t) of Eq. (19.10) (during the time of measurement) with respect to the information in the combined wave function ψ(x, y, t) of Eq. (19.5) [and Eq. (19.3)] (before the time of measurement).

From Eqs (19.5) and (19.3), ψ (x, y, t) is written as

 

Comparing Eq. (19.11b) with Eq. (19.10), it is clear that the state of the apparatus, which was f(y, t) before the interaction (with system S) now (after the interaction) may take various values f_m (y, t). In fact, according to Eq. (19.5) which expresses the total wave function before the interaction, the coefficient |a_m|² denotes the probability of the system S to have eigenvalue m of the operator . After interaction, the total wave function [Eq. (19.10)] gives same probability as |f_m(y, t)|². Therefore, the effect of interaction is that the information contained in the wave function of system S (in the form of |a_m|²) is transferred to the state of the apparatus (in the form of | f_m(y, t)|²).

The Schrodinger equation (19.9), with the form of the wave function of (19.10), becomes

Multiplying this equation by and integrating over x gives

This shows that the apparatus undergoes a change of state that is different for each eigenstate ϕ_n of the system operator . We suppose the interaction is strong enough so that the changes of the variables describing the apparatus will be so large that the state of the apparatus will depend mainly on the value n of the observable . It is, in fact, this correlation between two sets of states [state ϕ_m (of observable ) of system S and state f_m(y, t) of the apparatus] that is essential for making a measurement.

19.3  MEASUREMENT OF SPIN OF AN ATOM

After discussing the process of measurement for the general case, let us consider a particular example of measurement of components of the angular momentum of an atom (e.g., silver) with angular momentum Such an atom has completely filled shells with total orbital and spin angular momentum zero and one outermost electron in s-orbit with spin angular momentum So the atom has two possible states of spin angular momentum which are generally denoted by and

We shall use Stern–Gerlach experiment (described in Section 12.5) to measure spin angular momentum component S_z. As discussed in Section 12.5, the atomic beam passes through the inhomogeneous magnetic field. With the geometry shown in Figure 19.3, the atomic beam enters along y-axis and the magnetic field and its inhomogenity [i.e. the gradient (∂B_z/∂z)] are mainly in z-direction. The inhomogeneous magnetic field exerts a force on the atom in positive or negative z-direction according to whether the spin is up or down. We assume that the deflection of the atom by the inhomogeneous magnetic field is impulsive. Therefore, the motion of atom in z-direction, while it is moving in the region of inhomogeneous field may be neglected. However, as a result of the impulsive force on the atom (while in magnetic field), the atom shall have z-component of momentum in +z or –z direction, reaching the screen at a point above or below the central point.

Figure 19.3 (a) Stern–Gerlach arrangement (b) positions of the wave packets on the screen (schematic)

The interaction energy of atom and magnetic field may be written as

where is Pauli matrix and is Bohr magneton.

Now the beam is confined mainly in the y–z plane, whereas the magnetic field is mainly in z-direction. We may approximate the magnetic field as

 

where

So Eq. (19.14) becomes

Here measurement of z, the position of the atom, will give us information about the value of spin component S_z. Therefore, z is working as the apparatus coordinate. The combination of the inhomogeneous magnetic field, the coordinate z of the atom, and the detecting screen form our observing apparatus. Obviously [as clear from interaction Hamiltonian (19.17)], the function of inhomogeneous field B′ is to introduce correlation between atomic spin component and the apparatus coordinate z.

Before the atom enters the magnetic field, let its spin state be where a₊ and a_ are coefficients and and are spin wave functions corresponding to σ_z = + 1 and σ_z = – 1, respectively. Let the z-dependence of the atomic wave function be denoted by a wave packet f₀(z). So the initial wave function of the atomic system may be written as

When atom enters the field and interaction takes place [governed by the Hamiltonian Ĥ_I of Eq. (19.17)], the wavefunction becomes time-dependent given by Schrodinger equation

Here, the wave function ψ(z, t) may still be expanded in terms of spin wave functions χ₊ and χ_–, but now the expansion coefficients become function of z and t

similar to the case (of previous section) of going from the wave function (19.3) of the system before measurement to the wave function (19.10) during the time of measurement. The boundary condition dictates that at t = 0, the wave function (19.20) should be the same as wave function (19.18), so we must have:

 

and

 

Putting Eq. (19.20) in (19.19), we get

As we know

and

Therefore Eq. (19.22) gives

Now the coefficients of states and in Eq. (19.24) should be separately equal. Therefore, we have

and

These equations may be integrated using the boundary conditions (19.21) at t = 0, to give

 

 

The total wave function ψ(z, t) (19.20) may be written as

It is clear that the motion of the wave packet f₀(z) in z-direction is coupled to the spin state through the term . Therefore, the motion of the wave packet in z-direction has to be treated quantum mechanically. On the other hand, its motion along y-direction may be treated classically; that is, the error in finding time spent in the field region ∆t = (l/v) (l length of the field region, v = velocity in y-direction) may be considered to be very small (which is justified if l is large in comparison to the pole separation).

Now after the wave packet passes through the field, its wave function is given by Eq. (19.26) with t = ∆t = (l/v). Then we can see from Eq. (19.26) that the spin wave function is multiplied by the phase factor e^{–iµ_BzB′t/ħ} which simply means if spin is up, the momentum of the wave packet changes by while if the spin is down, the momentum changes by Depending upon the direction of momentum transfer the wave packet will move towards –ve z (for down spin state) or towards +ve z (for up spin state). [with the given geometry of magnetic poles in Fig. 19.3, the quantity B′ is –ve.] So it will be possible to measure the spin of atom by measuring the displacement of particle when it reaches the distant screen. The product B′∆t may be made large enough so as to obtain classically measurable separation between the packets corresponding to spin up and spin down states. In Figure 19.3(b), we show schematically the positions of the wave packets on the screen.

19.4  THE EPR PARADOX

We know that even though quantum mechanics tells only about the probability of a particular outcome in an experiment (whereas classical mechanics predicts the outcome of an experiment deterministically), still quantum mechanics remains the most successful theory as of now. In fact, quantum mechanics has never been shown to fail. Nevertheless, quantum mechanics has features which some physicists have found difficult to agree with Einstein, Podolsky and Rosen in 1935 raised questions on the validity of the generally accepted interpretations of quantum mechanics. Their criticism is based on their view that a complete physical theory should fulfill the following requirement:

1. ‘Every element of physical reality must have a counterpart in a complete physical theory’. They suggested the following (sufficient) criterion for recognizing an element of reality:
2.  ‘If, without in any way disturbing the system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to this physical quantity’.

Their objections to quantum mechanics were brought out clearly in a thought experiment, which we shall discuss below. The purpose of the experiment was to show that quantum mechanics, even though correct, does not confirm to simple requirement of physical reality together with another physically desirable requirement of separability or locality. Let us mention here that the locality condition, which derives from theory of relativity, states that no influence of any kind may propagate faster than the speed of light.

We now discuss the spin version (due to D. Bohm) of Einstein, Podolsky, and Rosen (EPR) thought experiment. Let us consider a molecule composed of two identical spin atoms in the singlet state, that is, S² = 0 eigenstate of the total spin operator Neglecting spin–orbit coupling, the spin part of the wave function of the two-atom system may be written as

Here and are spin up and spin down states, respectively, of atom 1 (with reference to z-axis). Similarly and are up and down spin states of atom 2. For two identical spin atoms in singlet state [Eq. (19.27a)], not only z-component of total spin is zero, but x- and y-components of total spin are zero. That means [S_z = (S_1z + S_2z) = 0, S_y = 0, S_z = 0]. We can easily check that the singlet wave function (19.27a) may be transformed (using Table 12.2) in terms of spin-up and spin-down states in (x- and y-) direction. For example, in terms of states with respect to x-direction, the wave function (19.27a) becomes [see Exercise (19.1)],

Similarly, in y-direction we get

Suppose the molecule is initially at rest and at t = 0 the molecule is disintegrated in such a way that the two atoms after separation move in opposite directions and their combined spin angular momentum remains zero. After separation there is no interaction between the atoms.

Suppose now we measures the z-component of spin of atom 1 with the help of Stern–Gerlach apparatus. We know the two-atom system is still in singlet spin state, therefore, we can immediately conclude that the z-component of spin of atom 2 is equal and opposite to that of atom 1. In fact, this would be the conclusion whether we use classical or quantum theory. However, there is a fundamental difference. In case of quantum mechanics, the fact remains that we can measure only one of the three components of the spin. Therefore, we can infer only about one component of spin of atom 2.

Now EPR argue that as

1. atoms 1 and 2 are very far from each other (and therefore non-interacting), the measurement on atom 1 is not going to affect atom 2, and
2. by measuring z-component of atom 1, we know about the z-component of atom 2 even without doing any experiment on atom 2 and therefore without in any way disturbing atom 2,

therefore, the z-component of atom 2 should be regarded as an element of reality existing separately in atom 2 alone. Further, according to EPR, this element of reality must have existed in atom 2 even before the measurement done on atom 1. But the observer of atom 1 could have chosen to re-orient his Stern–Gerlach apparatus in any arbitrary direction and got the value of spin component of atom 2 in any desired direction without in any way disturbing atom 2. This implies that the element of reality must exist in atom 2 corresponding to simultaneous definition of all three components of its spin. As the wave function can specify at most only one spin-component at a time with precision, the wave function does not provide a complete description of all the elements of physical reality existing in atom 2.

If the above line of reasoning is accepted, then there should be search for a new theory, which on one hand agrees with quantum mechanics and on the other hand gives a complete description of reality as defined and demanded above. This new theory may be called as local realistic theory.

19.5  THE HIDDEN VARIABLES AND BELL’S THEOREM

The local realistic theory to which EPR arguments lead, should be a theory which not only gives results what quantum mechanics gives but should also provide complete description of reality. This suggests that the new theory has additional (hidden) parameters which when averaged over reproduce quantum mechanics. The so called local hidden variables theory was considered by J. S. Bell in 1964 seriously.

Before discussing Bell’s local hidden variable theory, let us firstly discuss, following Bell, a slight modification of Bohm–EPR experiment. In this experiment, the first Stern–Gerlach set-up measures the spin component of atom 1 in z-direction, whereas the second set-up on the other side measures the spin component of atom 2 in a direction making an angle θ with z-axis (let us call this as z’-axis). For this, the second set-up has been rotated about x-axis through an angle θ. We can easily write the transformations connecting the up and down spin eigenstates in the rotated (z’-axis) and unrotated (z-axis) ones [see Section (12.6)]. The transformation is:

Putting this transformation in Eq. (19.27a), we have

So following usual statistical interpretation of the wave function, the probability that spin component of atom 1 is up in z-direction and spin component of atom 2 is up in z’-direction (at an angle θ with respect to z-axis) is

Let us now turn to Bell’s local hidden variables theory. Bell assumed that in addition to the wave function of the two spin system [Eq. (19.27a)], there is a set of hidden parameters λ = (λ₁, λ₂, ... λ_n) associated with the system (the two-spin system, i.e., diatomic molecule here). The set of parameters λ is different for each molecule. It is assumed that the value of λ, together with wave function , is sufficient for independently determining two components of spin of a single atom. When averaged over hidden variables λ, we get the usual quantum mechanical results. We shall not discuss detailed hidden variable theory here. Bell used the premises of local hidden variables theory and derived an inequality on various probabilities of the outcome of (Bohm–EPR) experimental results.

We follow here the simple method due to d’Espagnat to arrive at an equivalent form of Bell inequality. Following d’Espagnat we consider three directions denoted by A, B, and C. Let the spin state of atom 1 be measured in direction A and that of atom 2 in direction B. There may be four possible results: (A↑, B↑), (A↑, B↓), (A↓, B↑) (A↓, B↓). Let the corresponding fraction of pairs be denoted by P₁₂ (A↑, B↑) and so on. The Bell’s inequality, then may be expressed as

 

 

D’Espagnat used simple arguments of set theory to arrive at Bell’s inequality. As mentioned above, in local hidden variables theory, we assume that with the hidden parameters λ known for the molecule, it is possible to measure two components of spin of an atom contrary to the fact. Suppose this impossible instrument is able to measure spin components of one atom along A, B, and C directions. Let us determine (with this experiment) the spin components of a number of atoms in A and B directions. Of these atoms, there is a set of atoms which have spin component up in direction A and spin down in direction B. Let this set be designated by (A↑, B↓). For this set the spin component in third direction C is up for some atoms and down for the rest. So the set (A↑, B↓) divides itself into two parts: (A↑, B↓ C↑) and (A↑, B↓ C↓). If P(A↑, B↓) denotes the fraction of atoms having spin configuration (A↑, B↓), then we have

 

 

In a similar way, we get

 

 

From this equation, we may write

 

 

Similar arguments may be used to write

 

 

Using relations (19.33) in Eq. (19.32a), we find

 

 

It may be re-emphasized here that all the quantities and relations in Eqs (19.32a) to (19.34) have been written with the premise (of Bell) that with the set of hidden variables λ known, it may be possible to determine precisely the spin components of an atom in directions A, B, and C simultaneously. But we know in the usual experiments we can perform, it is not possible to measure all these components of spin of an atom. So it seems there is no way to test inequality Eq. (19.34) experimentally.

However, following EPR, if we use the assumption of separability, the probabilities of Eq. (19.34) for the measurements on one atom may be related to the measurements on pair of atom 1 and 2. If the measurements on atom 1 and 2 take place at far away places, the measurement on atom 2 is not going to affect atom 1. Now the singlet state has perfect negative correlations between the spin components of atom 1 and 2 (in any direction). Therefore, if atom 1 is found to have spin components (A↑, B↓) (in Bell measurement), then atom 2 must have components (A↓, B↑). Therefore, probability P(A↑, B↓) (of atom 1 having A↑ and B↓ spin-components) is same as probability P₁₂ (A↑, B↑) (of atom 1 having spin-component A↑ and atom 2 with spin-component B↑). Thus

With this equivalence, Eq. (19.31) follows directly from Eq. (19.34). This was the first result of local hidden variables theory in the form of Bell’s inequalities (in terms of quantities which are experimentally measurable) which is different than the corresponding results of quantum mechanics. And therefore, here is a chance to test the local realistic theory by checking whether experimental results honour Bell’s inequality or not.

During last three decades, there have been several experiments to test Bell’s inequality. In most of these experiments, the correlated photon pairs have been used. There are some elements, which when excited by laser light return to ground state in two steps, emitting a single photon in each step. The two photons thus emitted go in opposite directions and have opposite polarizations (i.e., opposite spins). The results have been found in agreement with quantum mechanics and negate Bell’s inequality.

The debate is going on as to which of the three premises used in EPR arguments and in local realistic (hidden variables) theory in deriving Bell’s inequality viz.

1. Criterion of reality
2. Criterion of locality or separability, and
3. Inductive inference

may be dispensed with.

19.6  TIME–EVOLUTION OF A SYSTEM: QUANTUM ZENO PARADOX

Let us consider an unstable quantum system (i.e., a system making transitions out of its initial state) and see what information do we get from the measurements on this system. There may be two categories of unstable quantum systems:

1. The systems which have discrete energy levels plus a continuum of energy levels; the system decays from its initial state (say ) to states including the continuum of states.
2. The systems which have a finite number of discrete energy levels; here the system decays from its initial state to finite number of final states.

Case I: In Section 16.3.1, we studied the transition of a system from its initial state to final states consisting of a discrete as well as continuum of states. We found that the probability P(t) that system has made transition away from initial state at time t is proportional to t. In Section 17.6, we found that the above mentioned observations led to the expression of survival probability P_a(t) (i.e., the probability that the system is still in its initial state ) at time t having exponentially decaying form

 

(with β = W_ba, the transition rate).

For small time, it gives

 

 

Let us divide time interval t in n small parts, then the non-decay probability, at the end of one part, will be

And at the end of interval ‘t’, the survival probability of initial state is

For observing the system continuously, we should have n → ∞. So

 

This simply means the system decays away from its initial state (to the continuum of states) with the exponential law, whether we observe it (at some interval) or not. So the process of (repeated) observations, here, is not going to affect the (exponential) decay law.

Case II: Let us next consider the unstable system, where it has finite number of discrete energy levels. Let the system be governed by the Hamiltonian

 

 

where unperturbed Hamiltonian Ĥ₀ has eigenstates given by

and V is the perturbing potential. Following the procedure mentioned in Chapter 16, the wave function (which is eigenstate of total Hamiltonian Ĥ) may be expressed as a linear combination of states

From Section 16.2, we have the equation governing time-development of coefficient C_n(t):

where

and

Let the system start evolving from state at t = 0, so we re-write Eq. (19.43) as

 

If the system is evolving slowly from state , the second term in Eq. (19.45) will not be significant (at least for small time t), so we have

Integrating it with respect to time gives

We have considered the case where once perturbation is switched on, it will be time independent. For small t, Eq. (19.47) gives

The probability of finding the system in state at time t is

So, if we make a measurement at time t to see if the system is still in its initial state , it will be found in with probability P_a(t) given by

For small t, it gives

If we divide time interval t in n small parts, the non-decay probability at the end of one part is

Thus, the survival probability of initial state, at the end of time t is

If, instead, the system is observed continuously, then we should have n → ∞. So

This simply means, the system does not decay at all in time interval t. So the continuous observations on unstable quantum system (having finite number of discrete energy levels) inhibits the system to evolve in time. Thus unstable systems under continuous observations shall not decay at all—is the result of quantum mechanics. This is called as Quantum Zeno Paradox.

That the result is not an artifact of the approximations made, may be seen by considering an exactly solvable two-level system, as follows.

Case III: Let a two-level system with eigenstates and and corresponding eigen energies and be perturbed by a time-dependent potential Ve^iωt. Then, the time-dependent wave function at time t may be written as

The time-dependent Schrodinger equation

 

with the wave function of Eq. (19.55) gives following equations for amplitudes C_a(t) and C_b(t):

With the substitution

 

 

equations (19.57) give

Let us start with the solution

 

 

Substituting Eqs (19.60) in Eqs (19.59), we get

The Eqs (19.61) have non-trivial solution if

which gives two values of ω₀, say ω_±, given by

where

So, we may write general solutions of Eqs (19.59) as

 

 

Using initial boundary conditions, C_a(0) = 1, C_b(0) = 0 or α_a(0) = 1, α_b(0) = 0, we have

 

 

So Eqs (19.64) give

 

 

Substituting these values in Eqs (19.59), we may easily get

Therefore,

 

The survival probability of the initial state given by | C_a(t) |² can be easily seen to be consisting of two cosine terms and obviously varies as t² for small times, just like that in Eq. (19.51). Therefore, again, we get the result of Eq. (19.54) for continuous observations.

EXERCISES

Exercise 19.1

Show that the spin singlet state of two spin particles written in the form of Eq. (19.27a) may be written in the forms (19.27b) and (19.27c).

Exercise 19.2

In a process an electron and a positron are emitted in +x and –x directions, respectively. The pair is emitted with zero spin–angular momentum and zero linear momentum. Let the spin be polarised along z direction. The pair is having total (kinetic) energy ħω. The electron is labelled as particle 1 and positron as 2.

• Write the total wave function of the pair including spin part, space part, and time dependence.
• What is the probability that a measurement gives the value of z-component of spin of positron as – (ħ/2)?
• If, instead, a measurement is done on positron to find its y-component, what is the probability that we get the value + (ħ/2)?
• Let a measurement of the z-component of spin of electron gives value (ħ/2). Write the wave function of the pair just after the measurement.
• After the above measurement (on electron), what will a measurement of the positron’s z-component of spin give?

Exercise 19.3

Consider the two spin particles in anti-symmetric singlet state given by Eq. (19.27a)

• Show that is an eigenstate of , , , and with eigenvalue zero.
  Here etc.
• Do you find it inconsistent that non-commuting operators and have simultaneous eigenstate ?

Exercise 19.4

Consider the two spin particles in the symmetric state given by

1. Show that is an eigenstate of and
2. Check if is an eigenstate of and .

SOLUTIONS

Solution 19.1

Using the ket and spinor notations of Table 12.2, we may write

Similarly,

Therefore,

which is same as Eq. (19.27b) (–ve sign does not matter). Similarly, we may get expression in terms of states with respect to y-direction.

Solution 19.2

• Let the electron be moving in +x direction and the positron in –x direction. With the given conditions, the space–time part of their wave functions may be written as

   

  e^{i(kx₁ – ωt)} and e^{– i (kx₂ – ωt)}

   

  where

  

  The total wave function of the spin–singlet pair may be written as

  
• The positron is equally likely to have values ±(ħ/2) of the z-component spin. So, probability for its value as –(ħ/2)
•  As seen in Exercise (19.1) the spin wave function in Eq. (19.27a) may be written with respect to spin state in y-direction and the form is same. So measurement of y-component of spin shall give ± (ħ/2) with equal probability.
• After the measurement the spin part collapses to so now total wave function is
  
• –(ħ/2)

Solution 19.3

1. Operator may be written as
   

   may be written in spinor form

   

   So

   

   Similarly, we may get operating and .

2. Refer to the Venn diagram in Figure 9.4 and related discussion

Solution 19.4

Operator may be written as

In the spinor form, state may be written as

•  
  

  so

  
  

REFERENCES

1. Bohm, D. 1951. Quantum Theory. New York: Prentice Hall.
2. d’Espagnat, B. 1979. ‘The Quantum Theory and Reality’, Scientific American, 241: 128.
3.  Einstein, A., Podolsky, B. and Rosen, N. 1935. ‘Can Quantum Mechanical Description of Physical Reality be Considered Complete?’, Phys. Rev., 47: 777.
4. Bell, J.S. 1964. ‘On the Einstein–Podolsky–Rosen Paradox’, Physics, 1: 195.
5. Merzbacher, E. 1999. Quantum Mechanics, 3rd edn., New York: John Wiley.
6.  Liboff, R.L. 1992. Introductory Quantum Mechanics. Mass: Addison-Wesley.
7.  Mermin, N.D. 1985. ‘Is the Moon There When Nobody Looks? Reality and the quantum Theory’, Physics Today, 38: 38.
8.  Deepak Kumar and Ishwar Singh, 1987. ‘Reality in Quantum Mechanics’, Physics Education, 3: 34. Ishwar Singh and Whitaker, M.A.B. 1982. ‘Role of the Observer in Quantum Mechanics and the Zeno Paradox’, Am. J. Phys., 50: 882.
9. Whitaker, M.A.B. and Ishwar Singh, 1982. ‘Interpretations of Quantum Mechanics and some Claimed Resolutions of the EPR Paradox’, J. Phys A., 15: 2377.
10. Whitaker, M.A.B. and Ishwar Singh, 1981. ‘Use of Reduced Density Matrix for the EPR Paradox’, Phys. Lett., 87A: 9.
11. Omnes, R. 1994. The Interpretation of Quantum Mechanics. New Jersey: Princeton University Press, Princeton.
12. Jammer, M. 1966. The Conceptual Development of Quantum Mechanics. New York: McGraw-Hill.