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2.3: Wavefunctions

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  • Consider a simple system with one classical degree of freedom, which corresponds to the Cartesian coordinate $ x$ . Suppose that $ x$ is free to take any value (e.g., $ x$ could be the position of a free particle). The classical dynamical variable $ x$ is represented in quantum mechanics as a linear Hermitian operator which is also called $ x$ . Moreover, the operator $ x$ possesses eigenvalues $ x'$ lying in the continuous range $ -\infty< x'<+\infty$ (since the eigenvalues correspond to all the possible results of a measurement of $ x$ ). We can span ket space using the suitably normalized eigenkets of $ x$ . An eigenket corresponding to the eigenvalue $ x'$ is denoted $ \vert x'\rangle$ . Moreover, [see Equation (85)]


    $\displaystyle \langle x' \vert x''\rangle = \delta(x'-x'').$ (117)



    The eigenkets satisfy the extremely useful relation [see Equation (87)]


    $\displaystyle \int_{-\infty}^{+\infty} d x' \, \vert x'\rangle\langle x'\vert= 1.$ (118)



    This formula expresses the fact that the eigenkets are complete, mutually orthogonal, and suitably normalized.

    A state ket $ \vert A\rangle$ (which represents a general state $ A$ of the system) can be expressed as a linear superposition of the eigenkets of the position operator using Equation (118). Thus,


    $\displaystyle \vert A\rangle = \int_{-\infty}^{+\infty} dx' \,\langle x'\vert A\rangle \vert x'\rangle$ (119)



    The quantity $ \langle x'\vert A\rangle$ is a complex function of the position eigenvalue $ x'$ . We can write


    $\displaystyle \langle x'\vert A\rangle = \psi_A(x').$ (120)



    Here, $ \psi_A(x')$ is the famous wavefunction of quantum mechanics. Note that state $ A$ is completely specified by its wavefunction $ \psi_A(x')$ [because the wavefunction can be used to reconstruct the state ket $ \vert A\rangle$ using Equation (119)]. It is clear that the wavefunction of state $ A$ is simply the collection of the weights of the corresponding state ket $ \vert A\rangle$ , when it is expanded in terms of the eigenkets of the position operator. Recall, from Section 1.10, that the probability of a measurement of a dynamical variable $ \xi$ yielding the result $ \xi'$ when the system is in state $ A$ is given by $ \vert\langle \xi'\vert A\rangle\vert^{\,2}$ , assuming that the eigenvalues of $ \xi$ are discrete. This result is easily generalized to dynamical variables possessing continuous eigenvalues. In fact, the probability of a measurement of $ x$ yielding a result lying in the range $ x'$ to $ x'+dx'$ when the system is in a state $ \vert A\rangle$ is $ \vert\langle x'\vert A\rangle\vert^{\,2}\,dx'$ . In other words, the probability of a measurement of position yielding a result in the range $ x'$ to $ x'+dx'$ when the wavefunction of the system is $ \psi_A(x')$ is


    $\displaystyle P(x', dx') = \vert\psi_A(x')\vert^{\,2}\, dx'.$ (121)



    This formula is only valid if the state ket $ \vert A\rangle$ is properly normalized: i.e., if $ \langle A\vert A\rangle = 1$ . The corresponding normalization for the wavefunction is


    $\displaystyle \int_{-\infty}^{+\infty} dx'\, \vert\psi_A(x')\vert^{\,2}= 1.$ (122)



    Consider a second state $ B$ represented by a state ket $ \vert B\rangle$ and a wavefunction $ \psi_B(x')$ . The inner product $ \langle B\vert A \rangle$ can be written


    $\displaystyle \langle B\vert A\rangle = \int_{-\infty}^{+\infty} dx'\,\langle B...
...\vert A \rangle = \int_{-\infty}^{+\infty} dx'\,\psi_B^\ast (x') \,\psi_A'(x'),$ (123)



    where use has been made of Equations (118) and (120). Thus, the inner product of two states is related to the overlap integral of their wavefunctions.

    Consider a general function $ f(x)$ of the observable $ x$ [e.g., $ f(x)=x^2$ ]. If $ \vert B\rangle = f(x)\,\vert A\rangle$ then it follows that


    $\displaystyle \psi_B(x') = \langle x'\vert f(x) \int_{-\infty}^{+\infty} dx''\,...
...nt_{-\infty}^{+\infty} dx''\, f(x'')\,\psi_A(x'') \,\langle x'\vert x''\rangle,$ (124)





    $\displaystyle \psi_B(x') = f(x')\, \psi_A(x'),$ (125)



    where use has been made of Equation (117). Here, $ f(x')$ is the same function of the position eigenvalue $ x'$ that $ f(x)$ is of the position operator $ x$ : i.e., if $ f(x)=x^2$ then $ f(x') = x'^{\,2}$ . It follows, from the above result, that a general state ket $ \vert A\rangle$ can be written


    $\displaystyle \vert A\rangle = \psi_A(x) \rangle,$ (126)



    where $ \psi_A(x)$ is the same function of the operator $ x$ that the wavefunction $ \psi_A(x')$ is of the position eigenvalue $ x'$ , and the ket $ \rangle$ has the wavefunction $ \psi(x') =1$ . The ket $ \rangle$ is termed the standard ket. The dual of the standard ket is termed the standard bra, and is denoted $ \langle$ . It is easily seen that


    $\displaystyle \langle \psi_A^{\,\ast}(x) \stackrel{\rm DC}{\longleftrightarrow} \psi_A(x)\rangle.$ (127)



    Note, finally, that $ \psi_A(x)\rangle$ is often shortened to $ \psi_A\rangle$ , leaving the dependence on the position operator $ x$ tacitly understood.