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Development Team Physics Paper No. : Quantum Mechanics-I Module : Generalized Uncertainty Principle Development Team Prof. Vinay Gupta, Department of Physics and Astrophysics, Principal Investigator University of Delhi, Delhi Prof. V. S. Bhasin, Professor , Department of Physics and Paper Coordinator Astrophysics, University of Delhi,Delhi Content Writer Prof. V. S. Bhasin, Professor , Department of Physics and Astrophysics, University of Delhi,Delhi Prof. Subash Chopra,Indian Institute of Technology, Delhi Content Reviewer 1 Quantum Mechanics-1 Physics Generalized Uncertainty Principle Description of Module Subject Name Physics Paper Name Quantum Mechanics-I Module Name/Title Generalized Uncertainty Principle Module Id M12 2 Quantum Mechanics-1 Physics Generalized Uncertainty Principle Contents of this Unit 1. Introduction 2. Generalized Uncertainty Principle 2.1 Root Mean Square Deviation 2.2 Derivation of the General Principle 2.2.1 Uncertainty relation between pair of canonically conjugate operators 2.2.2 Uncertainty Relation between the components of Angular Momentum Operators 3. Examples illustrating the use of Uncertainty Relations 4. States with Minimum Uncertainty Product 5. Time- Energy Uncertainty 6. Summary Learning Outcomes After studying this module, you shall be able to Know the definition of Root Mean Square Deviation and its significance Learn to deduce Generalized Heisenberg’s Uncertainty relationship for any pair of canonically conjugate variables corresponding to non-commuting operators Learn how this general relation is used in the case of (i) position and its conjugate momentum variables, (ii) between the components of orbital angular momentum Know the state, in coordinate representation, which corresponds to the minimum uncertainty product, Δx Δp= / 2 , in the position and momentum variables of a particle know.why the uncertainty relation in energy and time can not be considered analogous to that of the canonically conjugate position and momentum variables 3 Quantum Mechanics-1 Physics Generalized Uncertainty Principle 1.Introduction We have studied, in qualitative terms, the statement on Heisenberg’s uncertainty principle in terms of the uncertainties, ∆x and Δp in the measurement of position and momentum variables, the component of angular momentum and its canonically conjugate variable. Here we shall study how this principle is derived formally in general terms.by introducing the uncertainties ΔA and ΔB, associated with the measurement of the dynamical variables A and B in a given state. We shall use the generalized uncertainty relation to find out the uncertainty relation between the components of orbital angular momentum We shall also derive the expression for the wave packet which represents the minimum uncertainty in position and momentum variables. 2. Generalized Uncertainty Principle In module 3, we made the statement on Heisenberg’s uncertainty principle in terms of the uncertainties, ∆x and Δp in the measurement of position and momentum variables. Having studied the postulates of quantum mechanics, in particular, the commutation relation of quantum mechanical operators, we are now in a position to derive the generalized uncertainty principle general statement of the uncertainty principle for any pair of non- commuting Hermitian operators, Aˆ and Bˆ and use this in specific examples. 2.1 Root Mean Square Deviation As one of the postulates of quantum mechanics, we have learnt that an expectation value of an operator is an average over several measurements, each of which will deviate from the average value. The average of these deviations would of course be zero. But the average of the square of these deviations -- mean square deviations-- would be non-zero. The square root of the mean-square- deviation, called root mean square-deviation can be taken as measure of spread in the values measured . These spreads ΔA and ΔB in the measured values of the observed quantities A and B are called the uncertainties. 4 Quantum Mechanics-1 Physics Generalized Uncertainty Principle 2.2 Derivation of the General Statement Thus the uncertainties ΔA and ΔB associated with the measurement of A and B in a given quantum mechanical state , ψ, are defined as (A)2 (Aˆ Aˆ )2 { Aˆ 2 2 Aˆ Aˆ Aˆ 2} (12.1) Aˆ 2 Aˆ 2 The uncertainty depends on the state ψ in which the system is. Similarly, for the other Hermitian operator Bˆ , (B)2 (Bˆ Bˆ )2 Bˆ 2 Bˆ 2 (12.2) Let us define ˆ ˆ Eˆ Aˆ Aˆ and Fˆ Bˆ Bˆ A and B (12.3) and consider (Eˆ i Fˆ) (12.4) where, for simplicity, λ is real. Since Aˆ and Bˆ are Hermitian so are Eˆ and Fˆ. Now (Eˆ iFˆ )(Eˆ iFˆ ) 0 (12.5) or Eˆ 2 2Fˆ 2 i[Eˆ, Fˆ ] 0 where we note [Eˆ, Fˆ ] [(Aˆ Aˆ ),(Bˆ Bˆ )] [Aˆ, Bˆ] (12.6) Rewriting Eq.(12.5) in terms of the operators, , we have 2 2 2 (Aˆ Aˆ ) (Bˆ Bˆ ) i [Aˆ, Bˆ] 0 (12.7) 5 Quantum Mechanics-1 Physics Generalized Uncertainty Principle The left hand side of Eq.(12.7) has its minimum value when its derivative with respect to λ is zero. This gives 2(B)2 i [Aˆ, Bˆ] 0 [Aˆ, Bˆ] (12.8) or i 2(B)2 Substituting this value of λ from Eq.(12.8) in Eq.(12.7), we have 2 2 [Aˆ, Bˆ] [Aˆ, Bˆ] (Aˆ Aˆ 2) 0 (12.9) 4(B)2 2(B)2 Eq. (12.9) simplifies to 1 2 (A)2(B)2 [Aˆ, Bˆ] 4 Aˆ and Bˆ (12.10) 1 1 2 1 2 or [Aˆ, Bˆ] Cˆ 4 i 4 In Eq(12.10), we introduce the relation; [Aˆ, Bˆ] iCˆ , (12.11) where Cˆ is Hermitian.We get finally 1 (A) (B) Cˆ (12.12) 2 Eq.(12.12) gives the general statement of the uncertainty principle for any pair of non- commuting Hermitian operators, , which satisfy the relation Eq.(12.11). 2.2.1 Uncertainty relation between pair of canonically conjugate operators In particular, for a pair of conjugate operators , which satisfy the commutation relation [Aˆ, Bˆ] i , (12.13) we have, according to Eq.(12.12), 6 Quantum Mechanics-1 Physics Generalized Uncertainty Principle 1 (A) (B) (12.14) 2 And we know that a position coordinate variable x and its canonically conjugate momentum variable p satisfy the relation, Eq.(12.13). Accordingly, x p / 2 (12.15) This is the precise statement of the uncertainty principle in terms of the measurement of position and momentum observables, x and . The generalization to three dimensions is straight forward, i.e., 1 1 1 (x) (px) , (y) (py ) , and (z)(pz ) (12.16) 2 2 2 Corresponding to the commutation relation [xˆi , pˆ j] ii j i.j 1,2,3 (12.17) whereas we have [xˆi , xˆ j] 0 and [pˆi, pˆ j] 0 i.j 1,2,3 (12.18) 2.2.2 Uncertainty Relation between the components of Angular Momentum Operators We shall encounter in our future course of study several other examples of non-commuting Hermitian operators. Consider, for example, the orbital angular momentum of a particle, defined as; L r p about a fixed point. Expressed in Cartesian components, Lx ypz zp y, etc., in cyclic order Replacing the position and momentum variables as operators in quantum mechanics , we can write the commutation relations between its components, as for example, ˆ ˆ [Lx, Ly] [(yˆpˆz zˆpˆ y),(zˆpˆx xˆpˆz)] [yˆpˆz, zˆpˆx][yˆpˆz, xˆpz ][zˆpˆ y, zˆpˆx][zˆpˆ y, xˆpˆz ] Note that in the two middle terms, none of the operators are accompanied by its canonically conjugate partners. As a result, these terms would vanish. The first term can be expressed as 7 Quantum Mechanics-1 Physics Generalized Uncertainty Principle yˆpˆ x[ pˆ z, zˆ] i yˆpˆ x . Similarly, the last term gets simplified as [zˆ, pˆz]xˆpˆ y ixˆpˆ y. Putting these two terms together, we have ˆ ˆ ˆ [ Lx, Ly] i(xˆpˆ y yˆpˆx] iLz The commutators of other two components can be worked out in the same way. Thus we have ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [Lx.Ly] iLz , [Ly, Lz] iLx , [Lz, Lx] iLy (12.19) Applying the generalized uncertainty relation, Eq.(12.14) for the components of angular momentum operators we get (Lx)(Ly ) Lz (12.20) 2 i.e., the uncertainty in the product of measurement of the x- and y-components of orbital angular is determined by the expectation value of the z-component of the orbital momentum. Similar relations hold for other pairs of the angular momentum components as well. We shall be discussing in detail the quantal properties such as the eigenvectors and eigenvalues of angular momentum operators in the section on angular momentum. 3. Examples illustrating the use of Uncertaunty Relations Example 1: Let us consider a simple example of a linear harmonic oscillator to estimate, using the uncertainty relation Eq.(12.12), the minimum value of the zero point energy. We shall see that the uncertainty relation holds even for states in which the expectation values of are zero. Let ψ represent an eigenvector of belonging to the eigenvalue E and let f be respectively the operators corresponding to position and momentum, i.e., From Eq.(12.1) 2 1/ 2 x { xˆ } (12.21a) 8 Quantum Mechanics-1 Physics Generalized Uncertainty Principle Similarly, from Eq.(12.2) p { pˆ 2 }1/ 2 (12.21b) Since, in the case of oscillator, both are positive, we have x p / 2 So that (12.22) In fact according to Eq.(12.15), we have (12.15′) The important consequence of inequality (12.22) is that the lowest energy, called the zero point energy of the oscillator is different from zero. To make an estimate of this energy, we start from 1 1 E Hˆ pˆ 2 K xˆ2 2m 2 1 K (p)2 (x)2 (12.16) 2m 2 1 2K 1 (p)2 .
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