0
Math Prelude:
Orthogonal wave functions arise as a natural consequence of the mathematical structure of quantum mechanics and the relevant mathematical structure is called a Hilbert Space. Within this infinite dimensional (Hilbert) vector space is a definition of orthogonal that is exactly the same as "perpendicular" and that is the natural generalization of "perpendicular" vectors in ordinary three dimensional space. Within that context, wave functions are orthogonal or perpendicular when the "dot product" is zero.
Quantum Answer:
With that prelude, we can then say that mathematically, the collection of all quantum states of a quantum system defines a Hilbert Space. Two quantum functions in the space are said to be orthogonal when they are perpendicular and perpendicular means the "dot product" is zero.
Physics Answer:
The question asked has been answered, but what has not been answered (because it was not was not asked), is why orthogonal wave functions are important. As it turns out, anything that you can observe or measure about the state of a quantum system will be mathematically represented with Hermitian operators. A "pure" state, i.e. one where the same measurement always results in the same answers, is necessarily an eigenstate of a Hermtian operator and any two pure states that give two different results of measurement are necessarily "orthogonal wave functions."
Conclusion:
Thus, there are infinitely many orthogonal wave functions in the set of all wave functions of a quantum system and that orthogonal property has no physical meaning. When one identifies the subset of quantum states that associated pure quantum states (meaning specifically measured properties) and then two distinguishable measurement outcomes are associated with two different quantum states and those two are orthogonal. But, what was asked was a question of mathematics. Mathematically orthogonal wave functions do not guarantee distinct pure quantum state, but distinct pure quantum states does guarantee mathematically orthogonal wave functions. You can remember that in case someone asks.
Wiki User
∙ 10y ago
Add your answer:
Earn +20 pts
What is orthogonal and normalized wave function?
Everything you've ever wanted to know can be found at the link below.
What are the functions of the long arm of the chromosome?
To wave at chromosomes far away
What is normalising a wave function?
A wave function is normalized by determining normalization constants such that both the value and first derivatives of each segment of the wave function match at their intersections. If instead you meant renormalization, that is a different problem having to do with elimination of infinities in certain wave functions.
Is an electron a standing wave?
No -- an electron is a point particle with mass, charge, and spin. The probability that you will find an electron at a specific point can, however, often be calculated by wave functions. Any moving mass can be considered either a particle or a wave. Its properties can be defined via the deBorlie wave equation.
What is orthogonal?
It means "at right angles".
What is the definition of orthogonal signal space?
Orthogonal signal space is defined as the set of orthogonal functions, which are complete. In orthogonal vector space any vector can be represented by orthogonal vectors provided they are complete.Thus, in similar manner any signal can be represented by a set of orthogonal functions which are complete.
What has the author Richard Askey written?
Richard Askey has written: 'Three notes on orthogonal polynomials' -- subject(s): Orthogonal polynomials 'Recurrence relations, continued fractions, and orthogonal polynomials' -- subject(s): Continued fractions, Distribution (Probability theory), Orthogonal polynomials 'Orthogonal polynomials and special functions' -- subject(s): Orthogonal polynomials, Special Functions
What has the author Carl John Rees written?
Carl John Rees has written: 'Elliptic orthogonal polynomials' -- subject(s): Orthogonal Functions
What has the author James Ward Brown written?
James Ward Brown has written: 'Fourier series and boundary value problems' -- subject(s): Boundary value problems, Fourier series, Functions, Orthogonal, Orthogonal Functions 'Fourier series and boundary value problems' -- subject(s): Boundary value problems, Fourier series, Orthogonal Functions
What has the author David Leon Netzorg written?
David Leon Netzorg has written: 'Mechanical quadrature formulas and the distribution of zeros of orthogonal polynomials' -- subject(s): Orthogonal Functions
What is orthogonal and normalized wave function?
Everything you've ever wanted to know can be found at the link below.
What is meant by chemically orthogonal?
Chemically orthogonal means that two functional groups or molecules do not engage in similar/identical chemical reactions or exhibit significant differences in their chemical reactivities. An amino and a nitro are chemically orthogonal nitrogen-functions. On the other hand, an aldehyde and a ketone can be considered chemically very similar with respect to most reaction conditions. I have to admit that this terminology is rather general and does apply to many functions. It is probably not of too much use.
Which two trigonometric functions are used for analyzing vectors?
You usually need all three primary functions. The sine and cosine functions are used to resolve the vector along orthogonal axes, and the tangent function is used to find its direction.
Can the difference of 2 vectors be orthogonal?
The answer will depend on orthogonal to WHAT!
What are the functions of the tanks?
or studying wave properties
What has the author E I Peltola written?
E. I. Peltola has written: 'Comparison of some deuteron wave functions' -- subject(s): Deuterons, Wave functions
What is the orthogonal planning in ancient Greece?
it is planning of orthogonal planning
