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PHY4221Fall05_Notes2

PHY4221Fall05_Notes2 - PHY4221 Quantum Mechanics I Spring...

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PHY4221 Quantum Mechanics I Spring 2004 1.3 Representations In the preceding section we have developed the basic mathematical framework as used in Dirac’s formalism of quantum mechanics. We are now ready to apply the abstract theory to tackle real physical systems. To achieve this goal, it is sometimes more convenient to replace the abstract quantities by the more familiar mathematical structures and to work in terms of them. This procedure is similar to using coordinates in geometry, and has the advantage of giving one greater mathematical power for solving particular problems. The way in which the abstract quantities are to be replaced by more familiar mathematical structures is not unique, there being many possible ways corresponding to the many systems of coordinates one can have in geometry. Each of these ways is called a representation . When one has a particular problem to work out in quantum mechanics, one can minimize the labour by using a representation in which the problem becomes as simple as possible. 1.3.1 Momentum Operator in the Coordinate Space Consider the eigenvalue equation of the position operator ˆ q ˆ q | q i = q | q i (58) where | q i is the eigen ket vector corresponding to the eigenvalue q . Since the eigen- value q assumes a value between -∞ and , the position operator ˆ q is said to have a continuous eigen spectrum. The complete set of eigenvectors {| q i} are found to be mutually orthogonal and normalized to a Dirac delta function: h q 0 | q i = δ ( q 0 - q ). 1. Proof: In terms of the basis {| q i} , an arbitrary normalized ket vector | ψ i in the Hilbert space can be expressed as | ψ i = Z -∞ dq | q ih q | ψ i (59) 17
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for Z -∞ dq | q ih q | = ˆ I . (60) From Eq.(59), we obtain the following two equivalent expressions h ψ | ψ i = Z -∞ dq Z -∞ dq 0 h ψ | q 0 ih q 0 | q ih q | ψ i (61) and h ψ | ψ i = Z -∞ dq h ψ | q ih q | ψ i . (62) Comparing the above two different, yet equivalent, expressions of h ψ | ψ i yields h ψ | q i = Z -∞ dq 0 h ψ | q 0 ih q 0 | q i , (63) which in turn implies the result h q 0 | q i = δ ( q 0 - q ) . ( Q.E.D. ) (64) Next, let us introduce the operator ˆ T ² associated with shifting the eigenvalue q by a constant ² : ˆ T ² | q i = | q + ² i . (65) The action of this operator upon an arbitrary normalized ket vector | ψ i can be ex- pressed as ˆ T ² | ψ i = ˆ T ² Z -∞ dq | q ih q | ψ i = Z -∞ dq | q + ² ih q | ψ i = Z -∞ dq | q ih q - ² | ψ i (66) ˆ θ | ψ i ( lim ² 0 ˆ T ² - 1 ² ) | ψ i = Z -∞ dq | q i ( lim ² 0 h q - ² | ψ i - h q | ψ i ² ) = Z -∞ dq | q i ( - ∂q h q | ψ i ) . (67) 18
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From Eq.(67) we can easily obtain ˆ θ ˆ q - ˆ q ˆ θ · | ψ i = Z -∞ dq | q i ( - ∂q q h q | ψ i + q ∂q h q | ψ i ) = - Z -∞ dq | q ih q | ψ i = -| ψ i , (68) which implies that the operators ˆ θ and ˆ q obey the commutation relation: h ˆ θ, ˆ q i = - 1 . (69) Comparing this commutation relation with that of the operators ˆ q and ˆ p : [ˆ q, ˆ p ] = i ¯ h , we can identify the operator ˆ p as i ¯ h ˆ θ in the coordinate space: ˆ p i ¯ h ˆ θ = ¯ h i ∂q . (70) Accordingly, we can write ˆ q | ψ i = Z -∞ dq | q i { q h q | ψ i} ⇒ h q | ˆ q | ψ i = q h q |
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