Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

CRC Press eBooks

Wave Functions

Jonathan Allday

Published Mar 3, 2009• 0 citations
Publisher

Abstract

To a mathematician the funny ⊗ has a speci c meaning (they call it the tensor product). For us, it simply indicates that two independent aspects of the state have to be combined to get the full picture. In general terms one could write ⊗ S with S〉 being the spin state of the system and Ψ〉 the spatial state, the part that deals with energy, momentum, etc. e spin information is not contained within the spatial state; that has to be dealt with separately.1

DOI
10.1201/9781584887041-19

Related papers

Analysis of the spreading Gaussian wavepacket

Géza I. Márk

The ψ(x,t) wavefunction of a Gaussian wavepacket spreading in free space (V(x)≡0) is expressed in a didactic form. The expression found is a product of pure real factors and pure phase factors. This makes it very easy to derive the expression for the probability density from the wavefunction. The physical meaning of each of the factors is analysed. Zusammenfassen. Die Wellenfunktion ψ(x,t) eines Gaußschen Wellenpaketes, welches sich bei der Ausbreitung im freien Raum (V(x)≡0) verbreitert, wird in einer didaktischen Form ausgedrückt. Der gefundene Ausdruck ist ein Produkt von reellen Faktoren und reinen Phasen-Faktoren. Dies vereinfacht die Herleitung des Ausdruckes für die Wahrscheinlichkeitsdichte aus der Wellenfunktion. Die physikalische Bedeutung von jedem Faktor wird analysiert. While writing a paper [1] about the time evolution of different wavepackets I wanted to find a didactic expression for the ψ(x,t) wavefunction of a Gaussian initial state. The particular expression that was finally constructed is different from those found in quantum mechanics texts [2 ,3]. Figure 1. x- and t-dependent parts of the wavefunction. (a), (b) and (c) show the time development of the real part of factor 1 , factor 2 and of the full ψ(x,t). The x, y scale is the same for all (a), (b) and (c). (d) shows the time development of the probability density ρ(x,t). The x, y scale is the same for all time instants. Atomic units are used. a = 2.5 Bohr = 0.13 nm, nm. The atomic time unit is 2.41 × 10 -17 s. See the text for details. Figure 2. t-dependent parts of the wavefunction. (a) shows the time dependent prefactor of factor 2 as function of time. (b) and (c) show the time dependence of the terms of factor 3 . Their real (full curve) and imaginary (broken curves) parts are plotted against time. The thin dashed horizontal lines in (b) show the asymptotes for t= ∞ . Atomic units are used. See the text for details. Initial state Our initial state is a simple Gaussian wavepacket of the form This wavepacket is a product of three factors: A normalization factor that makes the norm of the wavefunction unity. A plane wave factor that accounts for the non-zero momentum p 0 of the wavepacket. A bell-shaped localizing function with half width at half maximum . Time evolution The time development of the initial ψ 0 (x) state is given by [ 2 ]: This Fourier integral can be calculated easily with Gaussian integrals and leads to a wavefunction like Transformation of into didactic form Now we want to transform this into something more informative. First note that the centre of the wavepacket is moving with the group velocity . Hence it is worth writing instead of x 0 into the first term of the numerator in the exponential. Working this out gives the following result It is getting clearer already! Now let us get rid of the complex denominators! Utilizing this we get finally where arg z is the phase of the complex number z, i.e. . Our ψ(x,t) has three main factors ((7), (8) and (9)). The first factor (7) is a product of two pure real coefficients and a plane wave. This plane wave part of factor 1 and the entire second (8) and third (9) factors are pure phase factors , i.e. their magnitude is one. Hence it is very easy to calculate the probability density ; one has only to calculate the square of the pure real coefficients of factor 1 which gives: The three terms of ψ(x,t) are as follows. Factor 1. (Cf (7)) A Gaussian of the form (1). This is an expression having the same form as ψ 0 (x) but the centre of gravity of the Gaussian is moving with speed and its width is increased to . The maximum value of the Gaussian is decreasing as its width increases making the area under ρ(x,t) (total probability) constant (one). The time evolution of factor 1 is shown in figure 1(a). Factor 2. (Cf (8)) An x- and t-dependent phase factor that is quadratic in x. One can see from figure 1(b) that this factor oscillates faster for larger |x| values. This accounts for the fact that the higher momentum components of the initial Gaussian ψ 0 (x) move with higher velocities. The function which describes the time-dependent prefactor of the phase is . This function (cf figure 2(a)) is not monotonic in time. Its value is zero for t = 0 and t= ∞ and has a maximum at . Factor 3. (Cf (9)) An x-independent (but still t-dependent) phase factor. This phase factor is a product of two terms. The first term is a monotonic function of time while the second one is oscillating. The phase of the first term is zero for t = 0 (a(t) is pure real) and -π/4 for t= ∞ (a(t) is pure imaginary). The second term is where and it accounts for the time development of the plane wave component in factor 1 . These two phase factors are plotted in figures 2(b) and 2(c) against time. Acknowledgment This work was partially supported by the Hungarian OTKA grant No F 014236. References [1] Márk G I Influence of the wavepacket shape to its time development to be published [2] Cohen-Tannoudji C, Diu B and Laloë F 1977 Quantum Mechanics (New York: Wiley) [3] Merzbacher E 1970 Quantum Mechanics 2nd edn (New York: Wiley)

European Journal of Physics•May 1, 1997•32 citations

CHAPTER 3. The Wavefunction and the Quantum State

Tim Maudlin

Princeton University Press eBooks•Mar 23, 2019•0 citations

The properties of wave tensors

Arthur Stanley Eddington

Abstract 1. In wave-mechanics there occur in addition to the ordinary physical vectors and tensors the four-valued quantities Ψ, Ψ introduced by Dirac. In this combination certain relations of invariance in regard to transformation appear which had escaped the ordinary tensor calculus. If we call the new type of quantity involving Ψ and its combinations Ψ-tensor, the position is that we cannot reach Ψ-tensors from the calculus of ordinary space tensors, but we can reach space tensors from a calculus of Ψ-tensors. I have shown that although Ψ-vectors cannot be expressed in terms of space vectors, mixed Ψ-tensors can be immediately resolved into space vectors. The present paper contains a simplification and systematisation of my earlier work on the Ψ-tensor calculus as well as a number of new results. The particular point round which the new results centre is as follows. The Ψ-tensors occurring in wave-mechanics are the product of two Ψ-vectors (Ψ, Ψ), and apparently the primary reason for introducing the unobservable quantity Ψ rather than working with physical tensors throughout is to impose this condition. I therefore examine the question: If we impose the condition that a wave tensor is the product of two Ψ-vectors, what is the corresponding restriction on the physical tensors equivalent to it ? The answer is (§7) that whatever is described by two Ψ-vectors Ψ, Ψ can equivalently be described by two space vectors of equal length at right angles to one another. One of these is the momentum vector; the other (generally ignored in current investigations) presumably represents positional relations (co-ordinates or distance); or rather I would regard it as the source of positional relations, which can only become explicit in more complicated developments involving many particles. The ordinary wave equation for one particle is obtained as an identity.

Proceedings of the Royal Society of London Series A Containing Papers of a Mathematical and Physical Character•Sep 1, 1931•4 citations

Spin and Wave Function of an Elementary Particle

P. Román

Nature•Sep 1, 1949•0 citations

About the Nature of the Wave Function and Its Dimensionality: The Case of Quantum Chemistry

Sebastián Fortín, Jesús Alberto Jaimes Arriaga

Synthese Library/Synthese library•Jan 1, 2022•5 citations

Problems

Oleg Boyarkin

Components of a field function are functions of coordinates and time and satisfy some definite differential equations, which are called field equations. Field equations can be given a very general variational or Lagrangian form, if one accepts, that a field as the dynamic system must be characterized by a definite Lagrangian function. For a mechanical system this function is written down as the sum over all the material points of the system. For a continuous system like a wave field, Lagrangian function is expressed by the space integral from Lagrangian function density: L(t) = ∫ Ld3x. (4.1) However, the variational principle deals not with the Lagrangian function but with an action S obtained from it by means of integration over x0 = ct S = c ∫ L(t)dt. (4.2) Therefore, noncovariant expression (4.1) in Lagrangian formalism is in fact intermediate and it is sufficient to consider the Lagrangian function density L, which depends on the four space-time coordinates x in an implicit way. Hereinafter, we shall simply call L by the Lagrangian. Analogously, the Hamiltonian function density H will be named the Hamiltonian. The Lagrangian depends on components of a field wavefunction ψk and on their first derivatives with respect to the space-time coordinates ∂ψk/∂x ν . In so doing we assign the meaning of generalized field coordinates to ψk, and the meaning of generalized field velocities to ∂ψk/∂x ν . The demand of lacking derivatives with respect to field functions above the first order in the Lagrangian is caused by the following reason. Dynamics of all known quantum fields is described either by equations of the first or the second order. On the one hand, these equations follow from the principle of a least action: δS = 0, (4.3) and under action variation the order of derivatives, entering L, is raised by one. The Lagrangian must be a relativistically covariant quantity, namely, a relativistic scalar L′(x′) = L(x). Thanks to the invariance under the space-time translations, we conclude that the Lagrangian must not manifestly contain any space-time coordinates x. Since dynamical observables (four-dimensional vector of the energy-momentum, the moment of momentum tensor, and so on) are real quantities, then the Lagrangian must be also a real quantity. We shall also impose the locality condition on the Lagrangian, that is, we assume that the Lagrangian depends only on the state of fields in the infinitely small vicinity of the point Particles, Fields, and Quantum x. So: L(x) = L(ψk, ∂ψk ∂xν ) = L (ψk(x), ψk;ν (x)) . General variation of an action, related with varying both field wave functions and of integration domain boundaries, is equal to: δS = δ ∫ L(x)dx = ∫ L(x′)dx′ − ∫ L(x)dx, (4.4) where dx = dx0dr. For L′(x′) we have the expression: L′(x′) = L (ψ′k(x′), ψ′k;ν(x′)) = L(x) + δL(x) = L(x) + δL(x) + dLdxλ δxλ, (4.5) In Eq. (4.5) we denote a form variation of corresponding quantities by the symbol δ. This way δL(x) represents varying the Lagrangian only on the account of the variation of field functions and field function derivatives: δL(x) = ∂L ∂ψk δψk + ∂L ∂ψk;ν δψk;ν , (4.6) and δψk(x) does a variation of a field function form: δψk(x) = ψ ′ k(x) − ψk(x).

Taylor & Francis eBooks•Feb 16, 2011•1 citation
To a mathematician the funny ⊗ has a speci c meaning (they call it the tensor product). For us, it simply indicates that two independent aspects of the state have to be combined to get the full picture. In general terms one could write ⊗ S with S〉 being the spin state of the system and Ψ〉 the spatial state, the part that deals with energy, momentum, etc. e spin information is not contained within the spatial state; that has to be dealt with separately.1
The ψ(x,t) wavefunction of a Gaussian wavepacket spreading in free space (V(x)≡0) is expressed in a didactic form. The expression found is a product of pure real factors and pure phase factors. This makes it very easy to derive the expression for the probability density from the wavefunction. The physical meaning of each of the factors is analysed. Zusammenfassen. Die Wellenfunktion ψ(x,t) eines Gaußschen Wellenpaketes, welches sich bei der Ausbreitung im freien Raum (V(x)≡0) verbreitert, wird in einer didaktischen Form ausgedrückt. Der gefundene Ausdruck ist ein Produkt von reellen Faktoren und reinen Phasen-Faktoren. Dies vereinfacht die Herleitung des Ausdruckes für die Wahrscheinlichkeitsdichte aus der Wellenfunktion. Die physikalische Bedeutung von jedem Faktor wird analysiert. While writing a paper [1] about the time evolution of different wavepackets I wanted to find a didactic expression for the ψ(x,t) wavefunction of a Gaussian initial state. The particular expression that was finally constructed is different from those found in quantum mechanics texts [2 ,3]. Figure 1. x- and t-dependent parts of the wavefunction. (a), (b) and (c) show the time development of the real part of factor 1 , factor 2 and of the full ψ(x,t). The x, y scale is the same for all (a), (b) and (c). (d) shows the time development of the probability density ρ(x,t). The x, y scale is the same for all time instants. Atomic units are used. a = 2.5 Bohr = 0.13 nm, nm. The atomic time unit is 2.41 × 10 -17 s. See the text for details. Figure 2. t-dependent parts of the wavefunction. (a) shows the time dependent prefactor of factor 2 as function of time. (b) and (c) show the time dependence of the terms of factor 3 . Their real (full curve) and imaginary (broken curves) parts are plotted against time. The thin dashed horizontal lines in (b) show the asymptotes for t= ∞ . Atomic units are used. See the text for details. Initial state Our initial state is a simple Gaussian wavepacket of the form This wavepacket is a product of three factors: A normalization factor that makes the norm of the wavefunction unity. A plane wave factor that accounts for the non-zero momentum p 0 of the wavepacket. A bell-shaped localizing function with half width at half maximum . Time evolution The time development of the initial ψ 0 (x) state is given by [ 2 ]: This Fourier integral can be calculated easily with Gaussian integrals and leads to a wavefunction like Transformation of into didactic form Now we want to transform this into something more informative. First note that the centre of the wavepacket is moving with the group velocity . Hence it is worth writing instead of x 0 into the first term of the numerator in the exponential. Working this out gives the following result It is getting clearer already! Now let us get rid of the complex denominators! Utilizing this we get finally where arg z is the phase of the complex number z, i.e. . Our ψ(x,t) has three main factors ((7), (8) and (9)). The first factor (7) is a product of two pure real coefficients and a plane wave. This plane wave part of factor 1 and the entire second (8) and third (9) factors are pure phase factors , i.e. their magnitude is one. Hence it is very easy to calculate the probability density ; one has only to calculate the square of the pure real coefficients of factor 1 which gives: The three terms of ψ(x,t) are as follows. Factor 1. (Cf (7)) A Gaussian of the form (1). This is an expression having the same form as ψ 0 (x) but the centre of gravity of the Gaussian is moving with speed and its width is increased to . The maximum value of the Gaussian is decreasing as its width increases making the area under ρ(x,t) (total probability) constant (one). The time evolution of factor 1 is shown in figure 1(a). Factor 2. (Cf (8)) An x- and t-dependent phase factor that is quadratic in x. One can see from figure 1(b) that this factor oscillates faster for larger |x| values. This accounts for the fact that the higher momentum components of the initial Gaussian ψ 0 (x) move with higher velocities. The function which describes the time-dependent prefactor of the phase is . This function (cf figure 2(a)) is not monotonic in time. Its value is zero for t = 0 and t= ∞ and has a maximum at . Factor 3. (Cf (9)) An x-independent (but still t-dependent) phase factor. This phase factor is a product of two terms. The first term is a monotonic function of time while the second one is oscillating. The phase of the first term is zero for t = 0 (a(t) is pure real) and -π/4 for t= ∞ (a(t) is pure imaginary). The second term is where and it accounts for the time development of the plane wave component in factor 1 . These two phase factors are plotted in figures 2(b) and 2(c) against time. Acknowledgment This work was partially supported by the Hungarian OTKA grant No F 014236. References [1] Márk G I Influence of the wavepacket shape to its time development to be published [2] Cohen-Tannoudji C, Diu B and Laloë F 1977 Quantum Mechanics (New York: Wiley) [3] Merzbacher E 1970 Quantum Mechanics 2nd edn (New York: Wiley)
Abstract 1. In wave-mechanics there occur in addition to the ordinary physical vectors and tensors the four-valued quantities Ψ, Ψ introduced by Dirac. In this combination certain relations of invariance in regard to transformation appear which had escaped the ordinary tensor calculus. If we call the new type of quantity involving Ψ and its combinations Ψ-tensor, the position is that we cannot reach Ψ-tensors from the calculus of ordinary space tensors, but we can reach space tensors from a calculus of Ψ-tensors. I have shown that although Ψ-vectors cannot be expressed in terms of space vectors, mixed Ψ-tensors can be immediately resolved into space vectors. The present paper contains a simplification and systematisation of my earlier work on the Ψ-tensor calculus as well as a number of new results. The particular point round which the new results centre is as follows. The Ψ-tensors occurring in wave-mechanics are the product of two Ψ-vectors (Ψ, Ψ), and apparently the primary reason for introducing the unobservable quantity Ψ rather than working with physical tensors throughout is to impose this condition. I therefore examine the question: If we impose the condition that a wave tensor is the product of two Ψ-vectors, what is the corresponding restriction on the physical tensors equivalent to it ? The answer is (§7) that whatever is described by two Ψ-vectors Ψ, Ψ can equivalently be described by two space vectors of equal length at right angles to one another. One of these is the momentum vector; the other (generally ignored in current investigations) presumably represents positional relations (co-ordinates or distance); or rather I would regard it as the source of positional relations, which can only become explicit in more complicated developments involving many particles. The ordinary wave equation for one particle is obtained as an identity.
Components of a field function are functions of coordinates and time and satisfy some definite differential equations, which are called field equations. Field equations can be given a very general variational or Lagrangian form, if one accepts, that a field as the dynamic system must be characterized by a definite Lagrangian function. For a mechanical system this function is written down as the sum over all the material points of the system. For a continuous system like a wave field, Lagrangian function is expressed by the space integral from Lagrangian function density: L(t) = ∫ Ld3x. (4.1) However, the variational principle deals not with the Lagrangian function but with an action S obtained from it by means of integration over x0 = ct S = c ∫ L(t)dt. (4.2) Therefore, noncovariant expression (4.1) in Lagrangian formalism is in fact intermediate and it is sufficient to consider the Lagrangian function density L, which depends on the four space-time coordinates x in an implicit way. Hereinafter, we shall simply call L by the Lagrangian. Analogously, the Hamiltonian function density H will be named the Hamiltonian. The Lagrangian depends on components of a field wavefunction ψk and on their first derivatives with respect to the space-time coordinates ∂ψk/∂x ν . In so doing we assign the meaning of generalized field coordinates to ψk, and the meaning of generalized field velocities to ∂ψk/∂x ν . The demand of lacking derivatives with respect to field functions above the first order in the Lagrangian is caused by the following reason. Dynamics of all known quantum fields is described either by equations of the first or the second order. On the one hand, these equations follow from the principle of a least action: δS = 0, (4.3) and under action variation the order of derivatives, entering L, is raised by one. The Lagrangian must be a relativistically covariant quantity, namely, a relativistic scalar L′(x′) = L(x). Thanks to the invariance under the space-time translations, we conclude that the Lagrangian must not manifestly contain any space-time coordinates x. Since dynamical observables (four-dimensional vector of the energy-momentum, the moment of momentum tensor, and so on) are real quantities, then the Lagrangian must be also a real quantity. We shall also impose the locality condition on the Lagrangian, that is, we assume that the Lagrangian depends only on the state of fields in the infinitely small vicinity of the point Particles, Fields, and Quantum x. So: L(x) = L(ψk, ∂ψk ∂xν ) = L (ψk(x), ψk;ν (x)) . General variation of an action, related with varying both field wave functions and of integration domain boundaries, is equal to: δS = δ ∫ L(x)dx = ∫ L(x′)dx′ − ∫ L(x)dx, (4.4) where dx = dx0dr. For L′(x′) we have the expression: L′(x′) = L (ψ′k(x′), ψ′k;ν(x′)) = L(x) + δL(x) = L(x) + δL(x) + dLdxλ δxλ, (4.5) In Eq. (4.5) we denote a form variation of corresponding quantities by the symbol δ. This way δL(x) represents varying the Lagrangian only on the account of the variation of field functions and field function derivatives: δL(x) = ∂L ∂ψk δψk + ∂L ∂ψk;ν δψk;ν , (4.6) and δψk(x) does a variation of a field function form: δψk(x) = ψ ′ k(x) − ψk(x).