Operators In Quantum Mechanics
Operators In Quantum Mechanics
In quantum mechanics we have something called hermitian operators and these are the same thing as linear operators. The postulates told us that things that you observe/measure in a system correspond to an hermitian operator. This means they correspond to observables such as position, momentum, spin, polarization etc... To start to understand what this means, we will need to start with looking at general operators.
Here are some key properties of linear operators
L on a vector A gives a new vector BL|A> = |B>
L on a vector multiplied by a constant is the same as the constant multiplied by the operator on the vector
L k|A> = kL|A> (Where k is a constant)
L on the sum of two vectors is the same as L acting on the two vectors and the adding the vectors after.
L( |A> + |B> ) = L|A> + L|B>
If we say L acts on A to give C
L|A> = |C>
Then we can write the inner product of B and C like this
<B [L|A>] = <B|C>
<B [L|A>] = <B|L|A> = LBA
This is know as the matrix element of L between the vectors B and A. If we are given 2 basis vectors |n> and |m>, we can record all of the matrix elements within the two vectors. This isn't very important and if you haven't studied matrices then you should not worry too much about this part (I have studied matrices and even I am not very comfortable with this).
Hermitian Operators
In quantum mechanics these are real quantities that you can measure.
1. <B|H|A> = <A|H|B>*
H operators are such that you do not have to change the operator if you complex conjugate the vectors around it.
2. <A|H|A> = <A|H|A>*
The left hand side of the equation is its own complex conjugate and so it must be a real quantity
Hermitian operators are any operators that when you put it between the same state/vector, you get something real.
HAB=HBA*
HMN=HNM* where m and n are basis vectors.
Operators act on vectors and changes the direction of them. There may be certain directions where if you apply the operator, it does not change the direction however. If you have an operator such as doubling the x component of the vector, vectors with only an x component will not change direction. These are called the eigenvectors of that the operator.
Take an eigenvector |E>
H|E> = K|E>
This means K is the eigenvalue of H. The operator H multiplies the eigenvector |E> by K when it acts on |E>
Another property of hermitian operators is that they have eigenvectors. This means they have corresponding eigenvalues. This must be the case as if hermitian operators did not have eigenvectors, they may not have eigenvalues and so when we measured a system we would not get values?
The eigenvalues of H have to be real. This is because the possible values of an observable are given by the eigenvalues of the operator and the measurements we take give real numbers (we are yet to measure an electron in an imaginary position).
It turns out the eigenvectors of hermitan operators are orthogonal
There are always D mutually orthogonal eigenvectors for H.
This means the eigenvectors of H form a basis. These are all unit length and so if we multiply the eigenvector by a constant we can make the eigenvalue = 1.
Hk|E> = |E> (where k is a constant)
The implication of this is that when H acts on a system, we will get a new vector (lets call it |N>). We can describe |N> as a sum of the eigenvectors of H. In the case of a coin, the eigenvectors of Q (the operator corresponding to measuring heads or tails on a coin) would be |H> and |T>. These are the eigenvectors because they are the states for which an observable Q has definite values for the "headness" of a coin.
So if we have a coin |C> and observe it,
Q|C> we will get a new state |N>.
We can describe this a sum of the eigenvectors of Q, |H> and |T>. Classically this makes no sense but if we did somehow make a coin that could behave quantum mechanically, describing N as something such as |H> + |T> would make perfect sense.
With the knowledge we now have on the complex vector space and hermitian operators, we will be able to look at our first system in quantum mechanics. It is likely that everything isn't fully clear as of now but as we look at polarization as a real example, things will become much clearer.
Thanks for reading. If you enjoyed this post or any of my others, follow and subscribe to my blog. Feel free to discuss anything related to this post or ask questions in the comments below.
Spooky Action At A Distance - Why The Universe May Not Be Real
The Birth Of Quantum Mechanics - The Ultraviolet Catastrophe
Schrödinger’s Kittens - The Boundary Between Quantum And Classical Mechanics
Did you see my previous post? Click the link below to check it out
The Complex Vector Space
HMN=HNM* where m and n are basis vectors.
Eigenvalues
One of our postulates was that the possible values of an observable are given by the eigenvalues of the operator. Let's now try to get an idea of what eigenvalues are.
Operators act on vectors and changes the direction of them. There may be certain directions where if you apply the operator, it does not change the direction however. If you have an operator such as doubling the x component of the vector, vectors with only an x component will not change direction. These are called the eigenvectors of that the operator.
Take an eigenvector |E>
H|E> = K|E>
This means K is the eigenvalue of H. The operator H multiplies the eigenvector |E> by K when it acts on |E>
Another property of hermitian operators is that they have eigenvectors. This means they have corresponding eigenvalues. This must be the case as if hermitian operators did not have eigenvectors, they may not have eigenvalues and so when we measured a system we would not get values?
The eigenvalues of H have to be real. This is because the possible values of an observable are given by the eigenvalues of the operator and the measurements we take give real numbers (we are yet to measure an electron in an imaginary position).
It turns out the eigenvectors of hermitan operators are orthogonal
There are always D mutually orthogonal eigenvectors for H.
This means the eigenvectors of H form a basis. These are all unit length and so if we multiply the eigenvector by a constant we can make the eigenvalue = 1.
Hk|E> = |E> (where k is a constant)
The implication of this is that when H acts on a system, we will get a new vector (lets call it |N>). We can describe |N> as a sum of the eigenvectors of H. In the case of a coin, the eigenvectors of Q (the operator corresponding to measuring heads or tails on a coin) would be |H> and |T>. These are the eigenvectors because they are the states for which an observable Q has definite values for the "headness" of a coin.
So if we have a coin |C> and observe it,
Q|C> we will get a new state |N>.
We can describe this a sum of the eigenvectors of Q, |H> and |T>. Classically this makes no sense but if we did somehow make a coin that could behave quantum mechanically, describing N as something such as |H> + |T> would make perfect sense.
With the knowledge we now have on the complex vector space and hermitian operators, we will be able to look at our first system in quantum mechanics. It is likely that everything isn't fully clear as of now but as we look at polarization as a real example, things will become much clearer.
Thanks for reading. If you enjoyed this post or any of my others, follow and subscribe to my blog. Feel free to discuss anything related to this post or ask questions in the comments below.
Check Out My Other Posts On Quantum Mechanics (link to all posts)
The Birth Of Quantum Mechanics - The Ultraviolet Catastrophe
Schrödinger’s Kittens - The Boundary Between Quantum And Classical Mechanics
Did you see my previous post? Click the link below to check it out
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