2

What is the physical interpretation of orthogonality? By taking integration over a function and we always solve it and get the answer zero. But what does it exactly mean in physics?

Qmechanic
  • 201,751
Sarthak Trivedi
  • 53
  • I did not exactly understand the integration part. We integrate function over some interval. Like sin(x) * sin(2x) over interval [-pi, pi]. – NonStandardModel Jan 20 '17 at 12:00
  • You may find the answers to this question helpful – By Symmetry Jan 20 '17 at 12:18
  • We can infer that two states for which a measurement off is certain to give two different results are orthogonal. This gives us some insight into the physical significance of orthogonal states. P.A.M. Dirac in "The Principles of Quantum Mechanics". – Frobenius Jan 20 '17 at 17:53

5 Answers5

5

From my perspective, two states are orthogonal if you have 0 probability to measure one on the states when the system is prepared in the other one.

Let us take a 3 levels system ${|\psi_1\rangle,|\psi_2\rangle,|\psi_3\rangle}$ and consider a state $|\psi\rangle=\sqrt{\frac{3}{10}}|\psi_1\rangle + \sqrt{\frac{7}{10}}|\psi_2\rangle$. The state is not orthogonal to $|\psi_1\rangle$ or $|\psi_2\rangle$, has a measure would give 30% and 70% of chance to find those states. By contrast, there is no way to obtain $|\psi_3\rangle$ through the measurement, corresponding to $|\psi\rangle$ and $|\psi_3\rangle$ being orthogonal.

Pen
  • 1,219
2

What is the physical interpretation of orthogonality?

Think of the $x$, $y$ and $z $ coordinate system you use every day. No matter what mathematical manipulation we do, we cannot express one direction in terms of the other two. They are orthogonal and linearly independent.

The physical result of orthogonality is that systems can be constructed, in which the components of that system have their individual distinctiveness preserved.

Another example are the spherical harmonics functions, which are a complete set of orthogonal functions. They can be considered as a mathematical representation of the physical fact that the set of electron orbitals around, say a hydrogen atom, will always retain their distinctive arrangement.

enter image description here

This is just one example of the physical consequences of orthogonality, since as the previous answers state, it means that the states are independent from each other, that is we can always tell them apart.

1

Mathematically speaking, orthogonality is derived from a scalar product. For finite dimensional vectors, the scalar product is the inner product as you know it. IN QM the vector spaces we consider are often of infinite dimension. The vectors are represented as functions, and the inner product is an integral.

Fundamentally it is the same thing though.

You are asking for the physical interpretation of orthogonality. For simplicity, consider a 2D space. If two vectors are orthogonal, you can project any other vector onto them, add the projected vectors together, and you end up with the original one.

It means that the projections in an orthogonal basis are really independent. You can change one, without affecting the other.

In my opinion, this is the crucial characteristic of orthogonal components. If you project the system onto one of them, change it, no other component will be affected.

polwel
  • 1,335
0

Very important aspect of orthogonal functions (on some interval) is that we can construct more complicated (periodic) functions as sum of simple functions. Like for example Fourier series sum of sin(nx) and cos(nx).

NonStandardModel
  • 224
-1

Orthogonal means that

  • sum of multiplication of two vectors results zero
  • or the integral of the multiple of two functions is zero.

Closely related terminology is: independent.

peterh
  • 8,208
  • This has nothing to do with physics. – Danu Jan 20 '17 at 13:08
  • I think the OP wants to know, in basic QM if for example we have two wave functions describing time evolution and they are orthogonal, as you say independent of each other, such as a Cos funtion cannot be described in Sin terms, what does that mean in a physical sense. –  Jan 20 '17 at 13:09
  • @Danu I would say it shows a lack of research effort, for sure, but physically I think its valid. –  Jan 20 '17 at 13:10
  • "sum of multiplication of two vectors"... Something is not right with this sentence. – Steeven Jan 20 '17 at 15:40
  • Thank you all of you. All of your answers cleared my concept about Orthogonalization. – Sarthak Trivedi Jan 23 '17 at 07:50