Is an axiom of Quantum Mechanics. The implications are extremely subtle and mathematical, so I recommend reading the quantum books by Shankar or Griffiths. Both should be available at any technical or university library.
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how does a scale operator make
The hamiltonian operator is the observable corresponding to the total energy of the system. As with all observables it is given by a hermitian or self adjoint operator. This is true whether the hamiltonian is limited to momentum or contains potential.
A binary operator is simply an operator that works with two operands (for example, two numbers). The binary operator is usually written between the two operands. Examples include the familiar operations of addition, subtraction, multiplication, or division - for example, in: 2 + 3 the "plus" is the binary operator, which works on the two numbers written on either side of it. What is an operator: Basically a function (calculation rule), written in a special way.
There is no "power" operator in C or C++. You need to the use the math library function pow().
And operator