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Phase Integral Approximations

Mark Sheard Child

Published Jan 3, 1991• 0 citations
Publisher

Abstract

Abstract The simplest representation for the quantum mechanical wavefunction in terms of classical quantities is due to Jeffreys (1925), Wentzel (1926), Kramers (1926), and Brillouin (1926a,b). It generalizes the free-motion form, ψ = exp(ipx/h), by allowing for variation in the momentum p(x) as a function of position, a correction that replaces the exponent by a phase integral, ∫p(x) dx, and introduces a normalization term related to the local velocity. A similar description may also be applied in classically forbidden regions by allowing the momentum to become imaginary, p(x) = i|p(x)|, provided that the local de Broglie wavelength, λ.(x) = h/p(x), varies sufficiently slowly with x. Details are given in Section 2.1 and discussed at greater length by Froman and Froman (1967) and Froman (1980).

DOI
10.1093/oso/9780198556541.003.0002

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Speculation on the Quantum Wavefunction in the Classical Limit

Francesco R. Ruggeri

Newtonian mechanics often employs a potential V(x) such that .5m v(x)v(x) + V(x) =E ((1)) where v(x) is velocity. Mathematically V(x) is defined at each x, but in reality there should be a tiny cutoff i.e. finite dx. Newtonian mechanics makes use of the idea of small dx lengths which tend to 0 mathematically. To first order v(x) is constant within a dx range, but the overall mathematics is continuous instead of discrete. Thus there is a continuous equation ((1)) for conservation of energy. A photon may be described by an eclectic field varying as cos(-Et+px). In terms of wavelengths exp(ipx) is often used as a mathematical convenience and noncontinous behaviour is not an issue. For example in refraction, momentum = hbar/wavelength abruptly changes and one straight line path is joined to another with a different slope. exp(ipx) is associated with repeatable wavelength patterns with a single wavelength representing a spatial distribution/ (kind of probability) region. A particle with rest mass may also be described by exp(ipx) which suggests a characteristic length hbar/p. Within this length (i.e at each x point which exist within hbar/p) the particle has p and pp/2m. Thus for a square well potential there is conservation of energy at each x point even though there is a wavelength in the problem. This is fine for a constant momentum, but what about a potential V(x) which implies p must change at each dx region? exp(ipx) is an eigenfunction of -id/dx which is linked to spatial invariance. It manifests itself physically by interference (e.g. two slits) which again involves non-smooth motion (i.e. one compares rays with completely different slopes). The characteristic length hbar/p is fixed and associated with a single p, yet momentum is supposed to change from one x point to the next. hbar/p does not fit within a single dx and this is a problem. Quantum mechanics remedies this issue by allowing for a superposition of exp(ipx)s i.e. W(x) = wavefunction = Sum over p a(p) exp(ipx) such that energy conservation is forced at each x. Thus one combines the idea of wavelength (which is physical) with energy conservation which is also physical. A consequence of this, however, is that -id/W/dx / W leads to a strange average momentum which is not the square root of -1/2m d/dx dW/dx/ W as well as discrete energies. A mathematical benefit is that one has a continuous differential equation, namely the time-independent Schrodinger equation. The question we ask is: What happens in the so-called classical limit? In the nonrelativstic case this leads to Newtonian mechanics in which there is no wavelength. How does the wavelength disappear so to speak? We suggest (as has been suggested often in the literature and textbooks) that the wavelengths in W(x) may become extremely tiny so as to not be detected easily in classical length scales. Mathematically the time-independent Schrodinger equation holds and there is a smooth W(x) at all points, but we suggest that this smoothness may not be necessary because it is not found in refracting light. If the wavelength is tiny compared to small dx regions in the classical limit, then many wavelengths of a given p fit into dx. Thus the idea of spatial invariance approximately holds within dx and there is no issue of the wavelength spilling out of the dx region as it does when it is comparable to the size of the system as in quantum bound problems. Thus within dx a single p with many (or at least several) wavelengths holds and there is a single p. The kinetic energy is pp/2m and the potential V(x) where x is in the center of dx. One may think of p changing abruptly (nonsmoothly) at the boundary of one dx with another such that conservation of energy holds. Thus one has p wavelengths which do not spill out of dx and give an approximate sense of spatial invariance together with energy conservation. The only loss is that of a smooth wavefunction, but it is not clear that such smoothness is a requirement. V(x) is physically associated with a limiting dx length (at which the concept of V(x) does not hold, but rather there are stochastic hits) and if one expands this length i.e. chooses a much bigger dx in keeping with classical scales, it seems that there should not be a problem. It as if V(x) delivers an impulse hit at each dx junction (like a series of different media in refraction).. This approach leads to the decoherence of wavelengths in the classical limit.

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On the Wentzel-Brillouin-Kramers Approximate Solution of the Wave Equation

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We take as our starting point the free particle classical action which may be written as A= -Et+px if t and x are treated as independent variables. It is possible to find two relationships linking v (velocity) to x and t from dA=0. The first is the classical wave result vp = E while the second is p=Ev where E=mo/sqrt(1-vv) (c=1). Furthermore one may write dA/dx partial = p ((1a)) and dA/dt partial = -E. ((1b)) This may be generalized to two eigenvalue differential equations, but this immediately begs the question: Why do this? We argue that Lorentz invariant form -Et+px represents scaling of both t and x. For example, p scales x and we argue there should be physical relevance to this. This physical relevance results in the wavelength proportional to 1/p. In other words, -Et+px represents more than velocity. The treatment of the scaling for a quantum wave and classical wave differ, we argue, in that the former keeps E,t and p,x separate while a classical wave does not. In other words pv=E applies to a classical wave which is described by cos(-Et+px). In other words, one takes ((1a)) and ((1b)) which separate E,t and p,x into two independent equations and takes account of scaling by combining the two (to include both x and t) i.e. creating the classical wave equation: 1/vv d/dt d/dt partial W(x,t) = d/dx d/dx partial W(x,t). We argue that quantum mechanics on the other hand keeps ((1a)) and ((1b)) separate. The velocity (relativistic) p=Ev is not a wave velocity. A solution to ((1a)) is exp(ipx) and to ((1b)) exp(-iEt). These are usually combined for a free particle to give exp(-iEt+ipx) and a wave is described with a phase velocity pv(phase)=E. We prefer to keep exp(-iEt) and exp(ipx) separate . For example, for two slit interference one does not make use of phase velocity or exp(-iEt). For a bound state one uses p=Ev, but a single free particle is represented by exp(ipx) in W(x)=wavefunction = Sum over p a(p)exp(ipx). exp(-iEn t) appears after one has found En, using the exp(ipx)’s in the sense that one does not include an exp(- i pp/2m t) for each exp(ipx). We argue that by separating E,t and p,x, the latter leads to a scheme for which there is “no time”. It is true that the particle moves at a speed x/t, but p=m1v1=m2v2 etc so p does not uniquely define velocity, rather it defines an impulse hit 2p = Integral F dt . Thus the scaling px which leads to an x, p only wave exp(ipx) is linked to an interaction. If the particle is moving freely and not interacting, then one has classical determinism i.e. v=constant = x/t. As soon as an interaction occurs, one changes from variables x,t which are linked to x, P conditional (x/p) and there is uncertainty in the position of x within a wavelength. The phase in exp(ipx) is only dependent on x not on the time the particle reaches x. exp(ipx) thus represents a probability and one must consider a sum of different possible outcomes (as in a two slit experiment) in order to describe an interaction. This is very different from the local interaction of classical mechanics. Nevertheless for a bound state one makes use of the two formalisms i.e. KE(x)= Sum over p pp/2m a(p)exp(ipx) / W(x) = KEclassical(x). We try to discuss some of these ideas in this note.

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Quantum Wavefunction and Fluctuation of Classical Action

Francesco R. Ruggeri

The classical action: Integral (0,T) dt1 L(v(t1), x(t1),t1) may be varied to find a stationary solution which yields Newton’s second law, a standard procedure in classical mechanics. Is it possible to obtain quantum mechanical motion from this same classical action? We argue that one may write the action for a relativistic/nonrelativistic particle moving at constant speed in terms of X and T i.e. v=velocity=constant=X/T. Varying with respect to X and T yields: d/dT Action = -E and d/dX Action = p. Thus, changing X by delta X yields a change in the action of p delta X which holds for all X because of the constant velocity. If one envisions a periodic function such as exp(i Action), this behaves as exp(ipx) in terms of -id/dX. Thus, there is an intrinsic wavelength related to 1/p. This follows directly from small changes of the Action. Alternatively, one may create a differential equation such as the Klein-Gordon or Schrodinger equation from classical conservation of energy-momentum using id/dT exp(iAction) = E exp(i Action) and -id/dX exp(iAction) = p exp(i Action). We note that exp(ipx) which describes the fluctuational behaviour of the classical Action may have positive and negative values showing that fluctuations may be both positive and negative These negative and positive fluctuations are, however, periodic so there is some systematic behaviour occurring, it seems.. We argue that exp(ipx) is statistical as its modulus is 1, indicating that a free particle has equal probability to be at any x point. For a system with a potential V(x), one may create an ensemble describing fluctuations of all possible p values i.e. W(x)=wavefunciton= Sum over p a(p) exp(ipx). The fluctuations are regular positive and negative, but there is different wavelength behaviour for each p i.e. 1/p so there is interference to yield an average fluctuation pattern i.e. the wavefunction.

•Jun 10, 2021•0 citations
Abstract The simplest representation for the quantum mechanical wavefunction in terms of classical quantities is due to Jeffreys (1925), Wentzel (1926), Kramers (1926), and Brillouin (1926a,b). It generalizes the free-motion form, ψ = exp(ipx/h), by allowing for variation in the momentum p(x) as a function of position, a correction that replaces the exponent by a phase integral, ∫p(x) dx, and introduces a normalization term related to the local velocity. A similar description may also be applied in classically forbidden regions by allowing the momentum to become imaginary, p(x) = i|p(x)|, provided that the local de Broglie wavelength, λ.(x) = h/p(x), varies sufficiently slowly with x. Details are given in Section 2.1 and discussed at greater length by Froman and Froman (1967) and Froman (1980).
Newtonian mechanics often employs a potential V(x) such that .5m v(x)v(x) + V(x) =E ((1)) where v(x) is velocity. Mathematically V(x) is defined at each x, but in reality there should be a tiny cutoff i.e. finite dx. Newtonian mechanics makes use of the idea of small dx lengths which tend to 0 mathematically. To first order v(x) is constant within a dx range, but the overall mathematics is continuous instead of discrete. Thus there is a continuous equation ((1)) for conservation of energy. A photon may be described by an eclectic field varying as cos(-Et+px). In terms of wavelengths exp(ipx) is often used as a mathematical convenience and noncontinous behaviour is not an issue. For example in refraction, momentum = hbar/wavelength abruptly changes and one straight line path is joined to another with a different slope. exp(ipx) is associated with repeatable wavelength patterns with a single wavelength representing a spatial distribution/ (kind of probability) region. A particle with rest mass may also be described by exp(ipx) which suggests a characteristic length hbar/p. Within this length (i.e at each x point which exist within hbar/p) the particle has p and pp/2m. Thus for a square well potential there is conservation of energy at each x point even though there is a wavelength in the problem. This is fine for a constant momentum, but what about a potential V(x) which implies p must change at each dx region? exp(ipx) is an eigenfunction of -id/dx which is linked to spatial invariance. It manifests itself physically by interference (e.g. two slits) which again involves non-smooth motion (i.e. one compares rays with completely different slopes). The characteristic length hbar/p is fixed and associated with a single p, yet momentum is supposed to change from one x point to the next. hbar/p does not fit within a single dx and this is a problem. Quantum mechanics remedies this issue by allowing for a superposition of exp(ipx)s i.e. W(x) = wavefunction = Sum over p a(p) exp(ipx) such that energy conservation is forced at each x. Thus one combines the idea of wavelength (which is physical) with energy conservation which is also physical. A consequence of this, however, is that -id/W/dx / W leads to a strange average momentum which is not the square root of -1/2m d/dx dW/dx/ W as well as discrete energies. A mathematical benefit is that one has a continuous differential equation, namely the time-independent Schrodinger equation. The question we ask is: What happens in the so-called classical limit? In the nonrelativstic case this leads to Newtonian mechanics in which there is no wavelength. How does the wavelength disappear so to speak? We suggest (as has been suggested often in the literature and textbooks) that the wavelengths in W(x) may become extremely tiny so as to not be detected easily in classical length scales. Mathematically the time-independent Schrodinger equation holds and there is a smooth W(x) at all points, but we suggest that this smoothness may not be necessary because it is not found in refracting light. If the wavelength is tiny compared to small dx regions in the classical limit, then many wavelengths of a given p fit into dx. Thus the idea of spatial invariance approximately holds within dx and there is no issue of the wavelength spilling out of the dx region as it does when it is comparable to the size of the system as in quantum bound problems. Thus within dx a single p with many (or at least several) wavelengths holds and there is a single p. The kinetic energy is pp/2m and the potential V(x) where x is in the center of dx. One may think of p changing abruptly (nonsmoothly) at the boundary of one dx with another such that conservation of energy holds. Thus one has p wavelengths which do not spill out of dx and give an approximate sense of spatial invariance together with energy conservation. The only loss is that of a smooth wavefunction, but it is not clear that such smoothness is a requirement. V(x) is physically associated with a limiting dx length (at which the concept of V(x) does not hold, but rather there are stochastic hits) and if one expands this length i.e. chooses a much bigger dx in keeping with classical scales, it seems that there should not be a problem. It as if V(x) delivers an impulse hit at each dx junction (like a series of different media in refraction).. This approach leads to the decoherence of wavelengths in the classical limit.
A discussion of the Wentzel-Brillouin-Kramers method of obtaining an approximate solution of the wave equation is given from the point of view that it forms a link between the quantum theory of Bohr and the new quantum mechanics. This becomes clear when one compares the probability distributions (see Figs. 2 and 3) and calculates the mean values rk−\stackrel{-}{{r}^{k}}rk− (see Table I). It is shown that for high quantum numbers these become the classical values. The method leads also, as shown by Zwaan, to quantization by the classical phase integrals with the use of half-integer quantum numbers. In the last section the method is applied to the charged shell atom model. It is shown that the condition of smooth joining of the wave function is practically equivalent to half-integer quantization of the sum of the inner and outer phase integrals. Of course there is no longer a sharp distinction between penetrating and non-penetrating orbits. The Land\'e formula for the doublet separations is derived. The value of ${\ensuremath{\psi}}^{2}(0)$ (0), which occurs in the calculation of the hyperfine separations in 2S^{2}S2S states, is also given.
We take as our starting point the free particle classical action which may be written as A= -Et+px if t and x are treated as independent variables. It is possible to find two relationships linking v (velocity) to x and t from dA=0. The first is the classical wave result vp = E while the second is p=Ev where E=mo/sqrt(1-vv) (c=1). Furthermore one may write dA/dx partial = p ((1a)) and dA/dt partial = -E. ((1b)) This may be generalized to two eigenvalue differential equations, but this immediately begs the question: Why do this? We argue that Lorentz invariant form -Et+px represents scaling of both t and x. For example, p scales x and we argue there should be physical relevance to this. This physical relevance results in the wavelength proportional to 1/p. In other words, -Et+px represents more than velocity. The treatment of the scaling for a quantum wave and classical wave differ, we argue, in that the former keeps E,t and p,x separate while a classical wave does not. In other words pv=E applies to a classical wave which is described by cos(-Et+px). In other words, one takes ((1a)) and ((1b)) which separate E,t and p,x into two independent equations and takes account of scaling by combining the two (to include both x and t) i.e. creating the classical wave equation: 1/vv d/dt d/dt partial W(x,t) = d/dx d/dx partial W(x,t). We argue that quantum mechanics on the other hand keeps ((1a)) and ((1b)) separate. The velocity (relativistic) p=Ev is not a wave velocity. A solution to ((1a)) is exp(ipx) and to ((1b)) exp(-iEt). These are usually combined for a free particle to give exp(-iEt+ipx) and a wave is described with a phase velocity pv(phase)=E. We prefer to keep exp(-iEt) and exp(ipx) separate . For example, for two slit interference one does not make use of phase velocity or exp(-iEt). For a bound state one uses p=Ev, but a single free particle is represented by exp(ipx) in W(x)=wavefunction = Sum over p a(p)exp(ipx). exp(-iEn t) appears after one has found En, using the exp(ipx)’s in the sense that one does not include an exp(- i pp/2m t) for each exp(ipx). We argue that by separating E,t and p,x, the latter leads to a scheme for which there is “no time”. It is true that the particle moves at a speed x/t, but p=m1v1=m2v2 etc so p does not uniquely define velocity, rather it defines an impulse hit 2p = Integral F dt . Thus the scaling px which leads to an x, p only wave exp(ipx) is linked to an interaction. If the particle is moving freely and not interacting, then one has classical determinism i.e. v=constant = x/t. As soon as an interaction occurs, one changes from variables x,t which are linked to x, P conditional (x/p) and there is uncertainty in the position of x within a wavelength. The phase in exp(ipx) is only dependent on x not on the time the particle reaches x. exp(ipx) thus represents a probability and one must consider a sum of different possible outcomes (as in a two slit experiment) in order to describe an interaction. This is very different from the local interaction of classical mechanics. Nevertheless for a bound state one makes use of the two formalisms i.e. KE(x)= Sum over p pp/2m a(p)exp(ipx) / W(x) = KEclassical(x). We try to discuss some of these ideas in this note.
In a previous note (1), we argued that W(x+dx)= W(x) exp(i dx ) where = - i d/dx lnW mimicking the behaviour of exp(ipx) for a free quantum particle. In this note, we consider that for the conditional probability P(p/x)= a(p) exp(ipx)/W(x), a shift from x to x+dx leads to: P(p/ x+dx) = P(p/x) exp(i dx (p- ) ) = P(p/x)(1+ idx (p- )). We note that in shifting from x to x+dx the overall sum of probability should not change, so P(p / x+dx)= P(p/x) + P(p/x) g(p,x). If one assumes that g(p,x) is a function of p alone, the simplest solution is p- . If one considers Sum over p pp/2m P(p/x+dx) = Sum over p pp/2m P(p/x) = delta KE ave then this should be equal to Force(x) dx. We also consider classical hydrodynamical equations and compare these with modified equations applying to the bound state quantum situation. The two sets of equations are very similar with density in the classical case being replaced with W(x)exp(-iEt), the wavefunction, and E energy or V(x) (potential) terms replaced with 2E or 2V (i.e. also 2Force). In a previous note, we had considered hydrodynamical equations with the restriction 2m=1 which is too strong. Here any m (mass) may be used.
The classical action: Integral (0,T) dt1 L(v(t1), x(t1),t1) may be varied to find a stationary solution which yields Newton’s second law, a standard procedure in classical mechanics. Is it possible to obtain quantum mechanical motion from this same classical action? We argue that one may write the action for a relativistic/nonrelativistic particle moving at constant speed in terms of X and T i.e. v=velocity=constant=X/T. Varying with respect to X and T yields: d/dT Action = -E and d/dX Action = p. Thus, changing X by delta X yields a change in the action of p delta X which holds for all X because of the constant velocity. If one envisions a periodic function such as exp(i Action), this behaves as exp(ipx) in terms of -id/dX. Thus, there is an intrinsic wavelength related to 1/p. This follows directly from small changes of the Action. Alternatively, one may create a differential equation such as the Klein-Gordon or Schrodinger equation from classical conservation of energy-momentum using id/dT exp(iAction) = E exp(i Action) and -id/dX exp(iAction) = p exp(i Action). We note that exp(ipx) which describes the fluctuational behaviour of the classical Action may have positive and negative values showing that fluctuations may be both positive and negative These negative and positive fluctuations are, however, periodic so there is some systematic behaviour occurring, it seems.. We argue that exp(ipx) is statistical as its modulus is 1, indicating that a free particle has equal probability to be at any x point. For a system with a potential V(x), one may create an ensemble describing fluctuations of all possible p values i.e. W(x)=wavefunciton= Sum over p a(p) exp(ipx). The fluctuations are regular positive and negative, but there is different wavelength behaviour for each p i.e. 1/p so there is interference to yield an average fluctuation pattern i.e. the wavefunction.