Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

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Fourier's theorerm, applied to one-dimensional wavefunctions, yields 

Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev(200)

Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev(201)

Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRevwhere Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev represents wavenumber. However, Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. Hence, we can also write 

Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev(202)

Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev(203)

whereWave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev  is the momentum-space equivalent to the real-space wavefunction Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

At this stage, it is convenient to introduce a useful function called the Dirac delta-function. This function, denotedWave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev was first devised by Paul Dirac, and has the following rather unusual properties: Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev  is zero for x ≠ 0, and is infinite at x = 0. However, the singularity at x = 0 is such that 

Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev(204)

The delta-function is an example of what is known as a generalized functioni.e., its value is not well-defined at all x, but its integral is well-defined. Consider the integral 

Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Since Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev is only non-zero infinitesimally close to x = 0, we can safely replace f(x) by f(0) in the above integral (assuming  f(x) is well behaved at x = 0), to give 

Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev(206)

where use has been made of Eq. (204). A simple generalization of this result yields 

Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev (207)

which can also be thought of as an alternative definition of a delta-function.

Suppose that  Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev It follows from Eqs. (203) and (207) that 

Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev         (208)

 Hence, Eq. (202) yields the important result 

Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev(209)

Similarly, 

Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev       (210)

 It turns out that we can just as well formulate quantum mechanics using momentum-space wavefunctions, Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev, as real-space wavefunctions, Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. The former scheme is known as the momentum representation of quantum mechanics. In the momentum representation, wavefunctions are the Fourier transforms of the equivalent real-space wavefunctions, and dynamical variables are represented by different operators. Furthermore, by analogy with Eq. (192), the expectation value of some operator O(p) takes the form 

 Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev  (211)

Consider momentum. We can write 

Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev  (212)

 where use has been made of Eq. (202). However, it follows from Eq. (210) that 

Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev(213)

 Hence, using Eq. (207), we obtain

Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev(214)

Evidently, momentum is represented by the operator P in the momentum representation. The above expression also strongly suggests [by comparison with Eq. (158)] that Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev can be interpreted as the probability density of a measurement of momentum yielding the value P at time t. It follows that Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev must satisfy an analogous normalization condition to Eq. (140): i.e.,  

Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev           (215)

Consider displacement. We can write

Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev  (216)

 Integration by parts yields 

Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev  (217)

Hence, making use of Eqs. (210) and (207), we obtain

Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev  (218)

Evidently, displacement is represented by the operator 

Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev  (219)

in the momentum representation.

Finally, let us consider the normalization of the momentum-space wavefunction Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. We have 

Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev  (220)

 Thus, it follows from Eqs. (207) and (210) that  

Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev  (221)

 Hence, if Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev is properly normalized [see Eq. (140)] then Wave Function Representation in Momentum Space - Quantum Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev, as defined in Eq. (203), is also properly normalized [see Eq. (215)].

The existence of the momentum representation illustrates an important point: i.e., that there are many different, but entirely equivalent, ways of mathematically formulating quantum mechanics. For instance, it is also possible to represent wavefunctions as row and column vectors, and dynamical variables as matrices which act upon these vectors.

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