Wednesday, May 2, 2012

The Wavefunction

File:QuantumHarmonicOscillatorAnimation.gif
Some trajectories of a harmonic oscillator (a ball attached to a spring) in classical mechanics (A-B) andquantum mechanics (C-H). In quantum mechanics (C-H), the ball has a wave function, which is shown with real partin blue and imaginary part in red. The trajectories C,D,E,F, (but not G or H) are examples of standing waves, (or "stationary states"). Each standing-wave frequency is proportional to a possible energy level of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have any energy.
The Wavefunction is at the heart of quantum mechanics in Physics, therefore at the heart of understanding our Universe, as we live in a quantum mechanical universe. Developed by Ernest Schrodinger in the 1920's based on Louis DeBroglie's contribution that Matter can behave as waves, this equation has lead to more quantum "weirdness" as well as scientific advances than any other. While practicing physicists use density matrices in calculations, Physics undergrads study this important function, then square it (as the Wavefunction has 2 parts: Real and imaginary) to find probabilities. From Wikipedia:

A wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of space and time. The laws of quantum mechanics (the Schrödinger equation) describe how the wave function evolves over time. The wave function behaves qualitatively like other waves, like water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. This explains the name "wave function", and gives rise to wave-particle duality.
The most common symbols for a wave function are ψ or Ψ (lower-case and capital psi).
Although ψ is a complex number, |ψ|2 is real, and corresponds to the probability density of finding a particle in a given place at a given time, if the particle's position is measured.
The SI units for ψ depend on the system. For one particle in three dimensions, its units are m–3/2. These unusual units are required so that an integral of |ψ|2 over a region of three-dimensional space is a unitless probability (i.e., the probability that the particle is in that region). For different numbers of particles and/or dimensions, the units may be different (though can be determined by dimensional analysis).
The wave function is absolutely central to quantum mechanics—it makes the subject what it is. It is also the source of the mysterious consequences and philosophical difficulties in what quantum mechanics means in nature, and even how nature itself behaves at the atomic scale and beyond—topics that continue to be debated today.

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[edit]Historical background

In the 1920s and 1930s, there were two divisions (so to speak) of theoretical physicists who simultaneously founded quantum mechanics: one for calculus and one for linear algebra. Those who used the techniques of calculus included Louis de Broglie, Erwin Schrödinger, Paul Dirac, Hermann Weyl, Oskar Klein, Walter Gordon, Douglas Hartree andVladimir Fock. This hand of quantum mechanics became known as "wave mechanics". Those who applied the methods of linear algebra included Werner Heisenberg, Max Born,Wolfgang Pauli and John Slater. This other hand of quantum mechanics came to be called "matrix mechanics". Schrödinger was one who subsequently showed that the two approaches were equivalent.[1] In each case, the wavefunction was at the centre of attention in two forms, giving quantum mechanics its unity.
De Broglie could be considered the founder of the wave model in 1925, due to his symmetric relation between momentum and wavelength: the De Broglie equation. Schrödinger searched for an equation that would describe these waves, and was the first to construct and publish an equation for which the wave function satisfied in 1926, based on classicalenergy conservation. Indeed it is now called the Schrödinger equation. However, no-one, even Schrödinger and De Broglie, were clear on how to interpret it. What did this functionmean? [2] Around 1924–27, Born, Heisenberg, Bohr and others provided the perspective of probability amplitude.[3] This is the Copenhagen interpretation of quantum mechanics. There are many other interpretations of quantum mechanics, but this is considered the most important - since quantum calculations can be understood.
In 1927, Hartree and Fock made the first step in an attempt to solve the N-body wave function, and developed the self-consistency cycle: an iterative algorithm to approximate the solution. Now it is also known as the Hartree–Fock method.[4] The Slater determinant and permanent (of a matrix) was part of the method, provided by Slater.
Interestingly, Schrödinger did encounter an equation for which the wave function satisfied relativistic energy conservation before he published the non-relativistic one, but it lead to unacceptable consequences for that time so he discarded it.[5] In 1927, Klein, Gorden and Fock also found it, but taking a step further: enmeshed the electromagnetic interactioninto it and proved it was Lorentz-invariant. De Broglie also arrived at exactly the same equation in 1928. This wave equation is now known most commonly as the Klein–Gordon equation.[6]
By 1928 Dirac deduced his equation from the first successful unified combination of special relativity and quantum mechanics to the electron - the Dirac equation. He found an unusual character of the wavefunction for this equation: it was not a single complex number, but a spinor.[4] Spin automatically entered into the properties of the wavefunction. Although there were problems, Dirac was capable of resolving them. Around the same time Weyl also found his relativistic equation, which also had spinor solutions. Later other wave equations were developed: see Relativistic wave equations for further information.

[edit]Mathematical introduction

[edit]Wavefunctions as multi-variable functions - analytical calculus formalism

Multivariable calculus and analysis (study of functions, change etc.) can be used to represent the wavefunction in a number of situations. Superficially, this formalism is simple to understand for the following reasons.
  • It is more directly intuitive to have probability amplitudes as functions of space and time. At every position and time coordinate, the probability amplitude has a value by direct calculation.
  • Functions can easily describe wave-like motion, using periodic functions, and Fourier analysis can be readily done.
  • Functions are easy to produce, visualize and interpret, due to the pictorial nature of the graph of a function (i.e. curves, contours, and surfaces). When the situation is in a high number of dimensions (say 3-d space) - it is possible to analyse the function in a lower dimensional slice (say a 2-d plane) or contour plots of the function to determine the behaviour of the system within that confined region.
Although these functions are continuous, they are not deterministic; rather, they are probability distributions. Perhaps oddly, this approach is not the most general way to represent probability amplitudes. The more advanced techniques use linear algebra (the study of vectors, matrices, etc.) and, more generally still, abstract algebra (algebraic structures, generalizations of Euclidean spaces etc.).

[edit]Wave functions as an abstract vector space - linear/abstract algebra formalism

The set of all possible wave functions (at any given time) forms an abstract mathematical vector space. Specifically, the entire wave function is treated as a single abstract vector:
\psi(\mathbf{r}) \leftrightarrow |\psi\rangle
where |\psi\rangle is a column vector written in bra-ket notation. The statement that "wave functions form an abstract vector space" simply means that it is possible to add together different wave functions, and multiply wave functions by complex numbers (see vector space for details). (Technically, because of the normalization condition, wave functions form aprojective space rather than an ordinary vector space.) This vector space is infinite-dimensional, because there is no finite set of functions which can be added together in various combinations to create every possible function. Also, it is a Hilbert space, because the inner product of wave functions \Psi_1(x) and \Psi_2(x) can be defined as
\langle \Psi_1 | \Psi_2 \rangle \equiv \int\limits_{-\infty}^\infty \Psi_1^*(x)\Psi_2(x) \, \mathrm{d}x,
where * denotes complex conjugate.
There are several advantages to understanding wave functions as elements of an abstract vector space:
  • All the powerful tools of linear algebra can be used to manipulate and understand wave functions. For example:
    • Linear algebra explains how a vector space can be given a basis, and then any vector can be expressed in this basis. This explains the relationship between a wave function in position space and a wave function in momentum space, and suggests that there are other possibilities too.
    • Bra-ket notation can be used to manipulate wave functions.
  • The idea that quantum states are vectors in a Hilbert space is completely general in all aspects of quantum mechanics and quantum field theory, whereas the idea that quantum states are complex-valued "wave" functions of space is only true in certain situations.

[edit]Introduction to vector formalism

Given an isolated physical system, the allowed states of this system (i.e. the states the system could occupy without violating the laws of physics) are part of a Hilbert space H. Some properties of such a space are
  • If | \psi \rangle and | \phi \rangle are two allowed states, then a | \psi \rangle + b | \phi \rangle\,\! is also an allowed state, provided |a|^2+|b|^2=1\,\!. (This condition is due to normalisation, see below.)
  • There is always an orthonormal basis of allowed states of the vector space H.
Physically, the nature of the inner product is dependent on the basis in use, because the basis is chosen to reflect the quantum state of the system.
When the basis is a countable set \{ | \phi_i \rangle \}\, and orthonormal, that is
\langle \phi_i | \phi_j \rangle = \delta_{ij},
then an arbitrary vector | \psi \rangle can be expressed as
| \psi \rangle = \sum_i c_i | \phi_i \rangle,
where the components are the (complex) numbers c_i = \langle \phi_i | \psi \rangle. This wave function is known as a discrete spectrum, since the bases are discrete.
When the basis is an uncountable set, the orthonormality condition holds similarly,
\langle \phi | \phi_0 \rangle = \delta \left ( \phi - \phi_0 \right ),
then an arbitrary vector | \psi \rangle can be expressed as
| \psi \rangle = \int c(\phi) | \phi \rangle \mathrm{d} \phi.
where the components are the functions  c(\phi) = \langle \phi | \psi \rangle. This wave function is known as a continuous spectrum, since the bases are continuous.
Paramount to the analysis is the Kronecker delta,  \delta_{ij}, and the Dirac delta function, \delta \left ( \phi - \phi_0 \right ) \,, since the bases used are orthonormal. More detailed discussion of wave functions as elements of vector spaces is below, following further definitions.

[edit]Requirements

Continuity of the wavefunction and its first spatial derivative (in the x direction, y and z coordinates not shown), at some time t.
The wavefunction must satisfy the following constraints for the calculations and physical interpretation to make sense:[7]
  • It must everywhere be finite.
  • It must everywhere be a continuous function, and continuously differentiable (at least up to all possible first order derivatives).
    • As a corollary, the function would be single-valued, else multiple probabilities occur at the same position and time, again unphysical.
  • It must everywhere satisfy the relevant normalization condition, so that the particle/system of particles exists somewhere with 100% certainty.
If these requirements are not met, it's not possible to interpret the wavefunction as a probability amplitude; the values of the wavefunction and its first order derivatives may not be finite and definite (with exactly one value), i.e. probabilities can beinfinite and multiple-valued at any one position and time - which is nonsense, as it does not satisfy the probability axioms. Furthermore, when using the wavefunction to calculate a measurable observable of the quantum system without meeting these requirements, there will not be finite or definite values to calculate from - in this case the observable can take a number of values and can be infinite. This is unphysical and not observed when measuring in an experiment. Hence a wavefunction is meaningful only if these conditions are satisfied.

[edit]Information about quantum systems

Main articles: Quantum state and Operator (physics)
Although the wavefunction contains information, it is a complex number valued quantity; only its relative phase and relative magnitude can be measured. It does not directly tell anything about the magnitudes or directions of measurable observables. An operator extracts this information by acting on the wavefunction ψ. For details and examples on how quantum mechanical operators act on the wave function, commutation of operators, and expectation values of operators; see Operator (physics).

[edit]Definition (single spin-0 particle in one spatial dimension)

[edit]Position-space wavefunction

For now, consider the simple case of a single particle, without spin, in one spatial dimension. (More general cases are discussed below). The state of such a particle is completely described by its wave function:
\Psi(x,t),
where x is position and t is time. This function is complex-valued, meaning that \Psi(x,t) is a complex number.
If the particle's position is measured, its location is not deterministic, but is described by a probability distribution. The probability that its position x will be in the interval [a, b] (meaning a ≤ x ≤ b) is:
P_{a<x<b} = \int\limits_a^b |\Psi(x,t)|^2 \,\mathrm{d}x
where t is the time at which the particle was measured. In other words, |\Psi(x,t)|^2 is the probability density that the particle is at x, rather than some other location.
This leads to the normalization condition:
\int\limits_{-\infty}^\infty |\Psi(x,t)|^2\, \mathrm{d}x = 1,
because if the particle is measured, there is 100% probability that it will be somewhere.

[edit]Momentum-space wavefunction

Main article: Momentum space
The particle also has a wave function in momentum space:
\Phi(p,t)
where p is the momentum in one dimension, which can be any value from -\infty to +\infty, and t is time. If the particle's momentum is measured, the result is not deterministic, but is described by a probability distribution:
P_{a<p<b} = \int\limits_a^b |\Phi(p,t)|^2 \mathrm{d}p,
analogous to the position case.
The normalization condition is also similar:
\int\limits_{-\infty}^{\infty} \left | \Phi \left ( p, t \right ) \right |^2 \mathrm{d}p = 1.

[edit]Relation between wavefunctions

The position-space and momentum-space wave functions are Fourier transforms of each other, therefore both contain the same information, and either one alone is sufficient to calculate any property of the particle. For one-dimension:[8]
\begin{align} \Phi(p,t) & = \frac{1}{\sqrt{2\pi\hbar}}\int\limits_{-\infty}^\infty e^{-ipx/\hbar} \Psi(x,t)\mathrm{d}x \\ &\upharpoonleft \downharpoonright\\ \Psi(x,t) & = \frac{1}{\sqrt{2\pi\hbar}}\int\limits_{-\infty}^\infty e^{ipx/\hbar} \Phi(p,t)\mathrm{d}p . \end{align}
Sometimes the wave-vector k is used in place of momentum p, since they are related by the de Broglie relation
p = \hbar k,
and the equivalent space is referred to as k-space. Again it makes no difference which is used since p and k are equivalent - up to a constant. In practice, the position-space wavefunction is used much more often than the momentum-space wavefunction.
Wave-particles "stationary".
Wave-particles travelling.
Interpretation of wave function for one spin-0 particle in one dimension. The wavefunctions shown are continuous, finite, single-valued and normalized. The colour opacity (%) of the particles corresponds to the probability density (which can measure in %) of finding the particle at the points on the x-axis.

[edit]Example of normalization

A particle is restricted to a 1D region between x = 0 and x = L; its wave function is:
\begin{align} \Psi (x,t) & = Ae^{i(kx-\omega t)}, & x \in [0,L] \\ \Psi (x,t) & = 0, & x \notin [0,L] \\ \end{align}
and is zero elsewhere. To normalize the wave function we need to find the value of the arbitrary constant A; solved from
\int\limits_{-\infty}^{\infty} |\Psi|^2 {\rm d}x = 1 .
From Ψ, we have |Ψ|2;
| \Psi | ^2 = A^2 e^{i(kx - \omega t)} e^{-i(kx - \omega t)} =A^2 ,
so the integral becomes;
\int\limits_{-\infty}^0 0 {\rm d}x + \int\limits_0^L A^2 {\rm d}x + \int\limits_L^\infty 0 {\rm d}x = 1 ,
therefore the constant is;
A^2 L = 1 \rightarrow A = \frac{1}{\sqrt{L}} .
The normalized wave function (in the region) is then given by;
\Psi (x,t) = \frac{1}{\sqrt{L}} e^{i(kx-\omega t)}, \quad x\in[0,L].

[edit]Definition (other cases)

[edit]Many spin-0 particles in one spatial dimension

The previous wavefunction can be generalized to incorporate N particles in one dimension:
\Psi(x_1,x_2,\cdots x_N, t),
The probability that particle 1 is in an x-interval R1 = [a1,b1] and particle 2 in interval R2 = [a2,b2] etc., up to particle N in interval RN = [aN,bN], all measured simultaneously at time t, is given by:
P_{x_1\in R_1,x_2\in R_2 \cdots x_N\in R_N} = \int\limits_{a_1}^{b_1} \mathrm{d}x_1 \int\limits_{a_2}^{b_2} \mathrm{d}x_2 \cdots \int\limits_{a_N}^{b_N} \mathrm{d}x_N | \Psi(x_1 \cdots x_N,t)|^2
The normalization condition becomes:
\int\limits_{-\infty}^\infty \mathrm{d}x_1 \int\limits_{-\infty}^\infty \mathrm{d}x_2 \cdots \int\limits_{-\infty}^\infty \mathrm{d}x_N |\Psi(x_1 \cdots x_N,t)|^2 = 1.
In each case, there are N one-dimensional integrals, one for each particle.
Visulization of the wavefunction for two spin-0 particles, in one dimension. Above: The two particles interact, then recoil from each other with 100% certainty. This situation occurs in quantum entanglement. Below: The particles are simply travelling.

[edit]One spin-0 particle in three spatial dimensions

[edit]Position space wavefunction

The electron probability density for the first few hydrogen atom electronorbitals shown as cross-sections. These orbitals form an orthonormal basis for the wave function of the electron. Different orbitals are depicted with different scale.
The position-space wave function of a single particle in three spatial dimensions is similar to the case of one spatial dimension above:
\Psi(\mathbf{r},t)
where r is the position in three-dimensional space (r is short for (x,y,z)), and t is time. As always \Psi(\mathbf{r},t)is a complex number. If the particle's position is measured at time t, the probability that it is in a region Ris:
P_{\mathbf{r}\in R} = \int\limits_R \left |\Psi(\mathbf{r},t) \right |^2 \mathrm{d}^3\mathbf{r}
(a three-dimensional integral over the region R, with differential volume element d3r, also written "dV" or "dx dy dz"). The normalization condition is:
\int\limits_{{\rm all \, space}} \left | \Psi(\mathbf{r},t)\right |^2 \mathrm{d}^3\mathbf{r} = 1,
where the integrals are taken over all of three-dimensional space (or 3d momentum space).

[edit]Momentum space wavefunction

There is a corresponding momentum space wavefunction for three-dimensions also:
\Phi(\mathbf{p},t)
where p is the momentum in 3-dimensional space, and t is time. This time there are three components of momentum which can have values -\infty  to + \infty  in each direction, in Cartesian coordinates x, y, z.
The probability of measuring the momentum components px between a and b, py between c and d, and pz between e and f, is given by:
P_{p_x\in[a,b],p_y\in[c,d],p_z\in[e,f]} = \int\limits_e^f \int\limits_c^d \int\limits_a^b \left | \Phi \left ( \mathbf{p}, t \right ) \right |^2 \mathrm{d}p_x \mathrm{d}p_y \mathrm{d}p_z ,
hence the normalization:
\int\limits_{{\rm all \, space}} \left | \Phi \left ( \mathbf{p}, t \right ) \right |^2 \mathrm{d}^3\mathbf{p} = 1.
analogous to space, d3p = dpxdpydpz is a differential 3-momentum volume element in momentum space.

[edit]Relation between wavefunctions

The generalization of the previous Fourier transform is [9]
\begin{align} \Phi(\mathbf{p},t) & = \frac{1}{\sqrt{\left(2\pi\hbar\right)^3}}\int\limits_{{\rm all \, space}} e^{-i \mathbf{r}\cdot \mathbf{p} /\hbar} \Psi(\mathbf{r},t)\mathrm{d}^3\mathbf{r} \\ &\upharpoonleft \downharpoonright\\ \Psi(\mathbf{r},t) & = \frac{1}{\sqrt{\left(2\pi\hbar\right)^3}}\int\limits_{{\rm all \, space}} e^{i \mathbf{r}\cdot \mathbf{p} /\hbar} \Phi(\mathbf{p},t)\mathrm{d}^3\mathbf{p}. \end{align}

[edit]Many spin-0 particles in three spatial dimensions

When there are many particles, in general there is only one wave function, not a separate wave function for each particle. The fact that one wave function describes many particles is what makes quantum entanglement and the EPR paradox possible. The position-space wave function for N particles is written:[4]
\Psi(\mathbf{r}_1,\mathbf{r}_2 \cdots \mathbf{r}_N,t)
where ri is the position of the ith particle in three-dimensional space, and t is time. If the particles' positions are all measured simultaneously at time t, the probability that particle 1 is in region R1 and particle 2 is in region R2 and so on is:
P_{\mathbf{r}_1\in R_1,\mathbf{r}_2\in R_2 \cdots \mathbf{r}_N\in R_N} = \int\limits_{R_1} \mathrm{d}^3\mathbf{r}_1 \int\limits_{R_2} \mathrm{d}^3\mathbf{r}_2\cdots \int\limits_{R_N} \mathrm{d}^3\mathbf{r}_N |\Psi(\mathbf{r}_1 \cdots \mathbf{r}_N,t)|^2
The normalization condition is:
\int\limits_{{\rm all \, space}} \mathrm{d}^3\mathbf{r}_1 \int\limits_{{\rm all \, space}} \mathrm{d}^3\mathbf{r}_2\cdots \int\limits_{{\rm all \, space}} \mathrm{d}^3\mathbf{r}_N |\Psi(\mathbf{r}_1 \cdots \mathbf{r}_N,t)|^2 = 1
(altogether, this is 3N one-dimensional integrals).
In quantum mechanics there is a fundamental distinction between identical particles and distinguishable particles. For example, any two electrons are fundamentally indistinguishable from each other; the laws of physics make it impossible to "stamp an identification number" on a certain electron to keep track of it.[10] This translates to a requirement on the wavefunction: For example, if particles 1 and 2 are indistinguishable, then:
\Psi \left (\mathbf{r},\mathbf{r'},\mathbf{r}_3,\mathbf{r}_4,\cdots \right ) = \pm \Psi \left ( \mathbf{r'},\mathbf{r},\mathbf{r}_3,\mathbf{r}_4,\cdots \right )
where the + sign is required if the particles are bosons, and the – sign is required if they are fermions. More exactly stated:
\Psi \left ( \mathbf{r},\mathbf{r'},\mathbf{r}_3,\mathbf{r}_4,\cdots \right ) = \left ( -1 \right )^{2s} \Psi \left ( \mathbf{r'},\mathbf{r},\mathbf{r}_3,\mathbf{r}_4,\cdots \right )
where s = spin quantum number,
integer for bosons: s \in \left \{ \pm 1,\pm 2,\pm 3 \cdots \right \},
and half-integer for fermions: s \in \left \{ \pm \frac{1}{2}, \pm \frac{3}{2} \cdots \right \} .
The wavefunction is said to be symmetric (no sign change) under boson interchange and antisymmetric (sign changes) under fermion interchange. This feature of the wavefunction is known as the Pauli principle.
For N interacting particles, i.e. particles which interact mutually and constitute a many-body system, the wavefunction is a function of all positions of the particles and time, it can't be separated into the separate wavefunctions of the particles. However, for non-interacting particles, i.e. particles which do not interact mutually and move independently, in a time-independent potential, the wavefunction can be separated into the product of separate wavefunctions for each particle:[7]
\Psi = \phi(t)\prod_{i=1}^N\psi(\bold{r}_i) = \phi(t)\psi(\bold{r}_1)\psi(\bold{r}_2)\cdots\psi(\bold{r}_N).

[edit]One particle with spin in three dimensions

Visulization of the wavefunction for a spin-1/2 particle, in one dimension. The spin orientations are shown in full opacity, with common notations for each value. Particles do not literally spin about their axes, this is just a representation.
For a particle with spin, the wave function can be written in "position-spin-space" as:
\Psi(\mathbf{r},s_z,t)
where r is a position in three-dimensional space, t is time, and sz is the spin projection quantum number along the z axis. (The z axis is an arbitrary choice; other axes can be used instead if the wave function is transformed appropriately, see below.) The sz parameter, unlike r and t, is a discrete variable. For example, for a spin-1/2 particle, sz can only be +1/2 or -1/2, and not any other value. (In general, for spin s, sz can be s, s–1,...,–s.) If the particle's position and spin is measured simultaneously at time t, the probability that its position is in R1 and its spin projection quantum number is a certain value mis:
P_{\mathbf{r}\in R,s_z=m} = \int\limits_{R} \mathrm{d}^3\mathbf{r} |\Psi(\mathbf{r},t,m)|^2
The normalization condition is:
\sum_{\mathrm{all\, }s_z} \int\limits_{{\rm all \, space}} |\Psi(\mathbf{r},t,s_z)|^2 \mathrm{d}^3\mathbf{r} = 1.
Since the spin quantum number has discrete values, it must be written as a sum rather than an integral, taken over all possible values.

[edit]Many particles with spin in three dimensions

Likewise, the wavefunction for N particles each with spin is:
\Psi(\mathbf{r}_1, \mathbf{r}_2 \cdots \mathbf{r}_N, s_{z\,1}, s_{z\,2} \cdots s_{z\,N}, t)
The probability that particle 1 is in region R1 with spin sz1 = m1 and particle 2 is in region R2 with spin sz2 = m2 etc. reads (probability subscripts now removed due to their great length):
P = \int\limits_{R_1} \mathrm{d}^3\mathbf{r}_1 \int\limits_{R_2} \mathrm{d}^3\mathbf{r}_2\cdots \int\limits_{R_N} \mathrm{d}^3\mathbf{r}_N \left | \Psi\left (\mathbf{r}_1 \cdots \mathbf{r}_N,m_1\cdots m_N,t \right ) \right |^2
The normalization condition is:
\sum_{s_{z\,N}} \cdots \sum_{s_{z\,2}} \sum_{s_{z\,1}} \int\limits_{{\rm all \, space}} \mathrm{d}^3\mathbf{r}_1 \int\limits_{{\rm all \, space}} \mathrm{d}^3\mathbf{r}_2\cdots \int\limits_{{\rm all \, space}} \mathrm{d}^3 \mathbf{r}_N \left | \Psi \left (\mathbf{r}_1 \cdots \mathbf{r}_N,s_{z\,1}\cdots s_{z\,N},t \right ) \right |^2 = 1
Now there are 3N one-dimensional integrals followed by N sums.
Again, for non-interacting particles in a time-independent potential the wavefunction is the product of separate wavefunctions for each particle:[7]
\Psi = \phi(t)\prod_{i=1}^N\psi(\bold{r}_i,s_{z\,i}) = \phi(t)\psi(\bold{r}_1,s_{z\,1})\psi(\bold{r}_2,s_{z\,2})\cdots\psi(\bold{r}_N,s_{z\,N}).

[edit]Normalization invariance

It is important that the properties associated with the wave function are invariant under normalization. If normalization of a wave function changed the properties, the process becomes pointless as we still cannot yield any information about the particle associated with the non-normalized wave function.
All properties of the particle, such as momentum, energy, expectation value of position, associated probability distributions etc., are solved from the Schrödinger equation (or otherrelativistic wave equations). The Schrödinger equation is a linear differential equation, so if Ψ is normalized and becomes AΨ (A is the normalization constant), then the equation reads:
\hat{H} (A\Psi) = i\hbar\frac{\partial }{\partial t}(A\Psi) \rightarrow \hat{H} \Psi = i\hbar\frac{\partial }{\partial t}\Psi
which is the original Schrödinger equation. That is to say, the Schrödinger equation is invariant under normalization, and consequently associated properties are unchanged.

[edit]Wavefunctions as vector spaces

Main article: Quantum state
As explained above, quantum states are always vectors in an abstract vector space (technically, a complex projective Hilbert space). For the wave functions above, the Hilbert space usually has not only infinite dimensions, but uncountably infinitely many dimensions. However, linear algebra is much simpler for finite-dimensional vector spaces. Therefore it is helpful to look at an example where the Hilbert space of wave functions is finite dimensional.

[edit]Basis representation

A wave function describes the state of a physical system  \left | \psi \right \rangle , by expanding it in terms of other possible states of the same system - collectively referred to as a basis orrepresentation  \left | \phi_i \right \rangle . In what follows, all wave functions are assumed to be normalized.
An element of a vector space can be expressed in different bases elements; and so the same applies to wave functions. The components of a wave function describing the same physical state take different complex values depending on the basis being used; however, just like elements of a vector space, the wave function itself is independent on the basis chosen. Choosing a new coordinate system does not change the vector itself, only the representation of the vector with respect to the new coordinate frame, since the componentswill be different but the linear combination of them still equals the vector.

[edit]Finite dimensional basis vectors

To start, consider the finite basis representation. A wave function \vec \psi with n components describes how to express the state of the physical system | \psi \rangle as the linear combination of nbasis elements | \phi_i \rangle, (i = 1, 2...n). The following is a breakdown of the used formalism.

[edit]Formalism

Conventional vector: Ψ and conventional notation
As a column vector or column matrix:
\vec \psi = \begin{bmatrix} c_1 \\ \vdots \\ c_n \end{bmatrix}.
State vector: Ψ and bra-ket notation
Equivalently in bra-ket notation, the state of a particle with wave function Ψ can be written as a ket;
\left | \psi \right \rangle = \sum_{i = 1}^n c_i \left | \phi_i \right \rangle = c_1 \left | \phi_1 \right \rangle + c_2 \left | \phi_2 \right \rangle + \cdots c_n \left | \phi_n \right \rangle = \begin{bmatrix} \left \langle \phi_1 \vert \psi \right \rangle \\ \vdots \\ \left \langle \phi_n \vert \psi \right \rangle \end{bmatrix} = \begin{bmatrix} c_1 \\ \vdots \\ c_n \end{bmatrix} .
The corresponding bra is the complex conjugate of the transposed matrix (into a row matrix/row vector):
\begin{align} \langle \psi | = | \psi \rangle^{*} & = \begin{bmatrix} \langle \phi_1 | \psi \rangle & \cdots & \langle \phi_n | \psi \rangle \end{bmatrix}^{*} = \begin{bmatrix} \langle \phi_1 | \psi \rangle^{*} & \cdots & \langle \phi_n | \psi \rangle^{*} \end{bmatrix} \\ & = \begin{bmatrix} c_1 & \cdots & c_n \end{bmatrix}^{*} = \begin{bmatrix} c_1^{*} & \cdots & c_n^{*} \end{bmatrix} \end{align} \,\!
By "the state of a particle with wavefunction Ψ", written as  | \psi \rangle \,\!, this means the variables which characterize the system, with respect to the wavefunction. The wave function associated with a particular state may be seen as an expansion of the state in a basis of H. For example, a basis could be for a free particle travelling in one dimension, with momentum eigenstates ψ± corresponding to the ±x direction:
| \psi \rangle = \psi_+ | p_{+} \rangle + \psi_{-} | p_{-} \rangle .
Another example is the superposition of two energy eigenstates for a particle trapped in a 1-d box (these states are stationary state):
| \psi \rangle = \psi_1 | E_1 \rangle + \psi_2 | E_2 \rangle.
The most characteristic example is a particle in a spin up or down configuration:
| \psi \rangle = \psi_+ | \uparrow_z \rangle + \psi_{-} | \downarrow_z \rangle ,
(see below for details of this frequent case). Notice how kets are not completely analogous to the ordinary notion of vectors - rather they are labels for a state of a wavefunction, which are used in a similar way. In all of the above examples, the particle is not in any one definite or preferred state, but rather in both at the same time - hence the termsuperposition. The free particle could be have momentum in the +x or –x direction simultaneously, the trapped particle in the 1-d potential well can be in the energy eigenstates corresponding to eigenvalues E1 and E2 at the same time, the particle with spin could be in spin up or down orientation at any instant of time. The relative chance of which state occurs is related to the (moduli squares of the) coefficients.

[edit]Infinite dimensional basis vectors

The case of a countably infinite vector, with a discrete index, is treated and interpreted in the same manner as a finite vector, except the sum is extended over an infinite number of basis elements.
Conventional vector: Ψ and conventional notation
As a column vector or column matrix, there are infinitely many entries:
\vec \psi = \begin{bmatrix} c_1 \\ \vdots \\ c_n \\ \vdots \end{bmatrix}.
State vector: Ψ and bra-ket notation
In bra-ket notation;
\left | \psi \right \rangle = \sum_{i = 1}^\infty c_i \left | \phi_i \right \rangle = c_1 \left | \phi_1 \right \rangle + c_2 \left | \phi_2 \right \rangle + \cdots = \begin{bmatrix} \left \langle \phi_1 \vert \psi \right \rangle \\ \vdots \\ \left \langle \phi_n \vert \psi \right \rangle \\ \vdots \end{bmatrix} = \begin{bmatrix} c_1 \\ \vdots \\ c_n \\ \vdots \end{bmatrix} .
The corresponding bra is as before:
\begin{align} \langle \psi | = | \psi \rangle^{*} & = \begin{bmatrix} \langle \phi_1 | \psi \rangle & \cdots & \langle \phi_n | \psi \rangle & \cdots \end{bmatrix}^{*} = \begin{bmatrix} \langle \phi_1 | \psi \rangle^{*} & \cdots & \langle \phi_n | \psi \rangle^{*} & \cdots \end{bmatrix} \\ & = \begin{bmatrix} c_1 & \cdots & c_n & \cdots \end{bmatrix}^{*} = \begin{bmatrix} c_1^{*} & \cdots & c_n^{*} & \cdots \end{bmatrix} \end{align} \,\!

[edit]Continuously indexed vectors

Now consider an uncountably infinite number of components of the physical state of the particle,  \left | \psi \right \rangle. For this reason the collection of all states  \left | \psi \right \rangle is known as a continuum orspectrum of states. Finite or countably infinite basis vectors are summed over a discrete index - for a continuous basis the integral is over the continuous index, replacing the sum.
Continuously indexed vector: Ψ and bra-ket notation
As usual  | \psi \rangle  is the physical state of the particle. The sum for a superposition of states now becomes an integral. In what follows, all integrals are with respect to the basis variable ϕ, over the required range. Usually this is just the real line or subset intervals of it. The state  | \psi \rangle  is given by:
| \psi \rangle = \int | \phi \rangle \langle \phi | \psi \rangle \mathrm{d}\phi = \int | \phi \rangle c(\phi) \mathrm{d}\phi = \int| \phi \rangle \psi(\phi) \mathrm{d}\phi .
See below for more on notation of basis and components.

[edit]Ontology

Main article: Interpretations of quantum mechanics
Whether the wave function really exists, and what it represents, are major questions in the interpretation of quantum mechanics. Many famous physicists of a previous generation puzzled over this problem, such as Schrödinger, Einstein and Bohr. Some advocate formulations or variants of the Copenhagen interpretation (e.g. Bohr, Wigner and von Neumann) while others, such as Wheeler or Jaynes, take the more classical approach[14] and regard the wave function as representing information in the mind of the observer, i.e. a measure of our knowledge of reality. Some, ranging from Schrödinger, Einstein, Bohm and Everett and others, argued that the wave function must have an objective, physical existence. The later argument was recently supported by the demonstration (not peer reviewed) of a theorem stating the physical reality of the quantum state.[15] For more on this topic, seeInterpretations of quantum mechanics.

[edit]Examples

Here are examples of wavefunctions for specific applications:

[edit]See also

[edit]References

  1. ^ Hanle, P.A. (December 1977), "Erwin Schrodinger's Reaction to Louis de Broglie's Thesis on the Quantum Theory.", Isis 68 (4): 606–609, doi:10.1086/351880
  2. ^ Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, ISBN 0-7167-8964-7
  3. ^ Sears' and Zemansky's University Physics, Young and Freedman (12th edition), Pearson Ed. & Addison-Wesley Inc., 2008, ISBN 978-0-321-50130-1
  4. ^ a b c Quanta: A handbook of concepts, P.W. Atkins, Oxford University Press, 1974, ISBN 0-19-855493-1
  5. ^ Quantum Field Theory, D. McMahon, Mc Graw Hill (USA), 2008, ISBN 978-0-07-154382-8
  6. ^ Particle Physics (3rd Edition), B.R. Martin, G. Shaw, Manchester Physics Series, John Wiley & Sons, 2008, ISBN 978-0-470-03294-7
  7. ^ a b c Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0
  8. ^ Griffiths, page 107 of the first edition
  9. ^ Quantum Mechanics (3rd Edition), Eugen Merzbacher, 1998, John Wiley & Sons, ISBN 0-471-88702-1
  10. ^ Griffiths, p179 of the first edition
  11. ^ http://www.users.csbsju.edu/~frioux/dirac/dirac.pdf , F. Rioux
  12. ^ Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0
  13. ^ http://www.nyu.edu/classes/tuckerman/adv.chem/lectures/lecture_9/node2.html
  14. ^ E. T. Jaynes. Probability Theory: The Logic of Science, Cambridge University Press (2003),
  15. ^ Pusey, Matthew F.; Jonathan Barrett, Terry Rudolph (14). "The quantum state cannot be interpreted statistically". arXiv.org: arxiv:1111.3328v1.
2.Quantum Mechanics(Non-Relativistic Theory), L.D. Landau and E.M. Lifshitz, ISBN 0-08-020940-8

[edit]Further reading

[edit]External links



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