bon said:Homework Statement
Suppose we have a system and that {|a>, |b>, ...} is a complete and orthonormal basis for the system
Am i right in thinking Ʃ(j) <k|j><j|i> = <k|i> = 0 unless k=i?
In other words, does the LHS expression equal the middle one because Ʃ(j) |j><j| is just the insertion of the identity and we can put it in anywhere?
Homework Equations
The Attempt at a Solution
I've explained my attempt above.
Thanks!
Dick said:You need to say what |k> and |i> are. If they are elements of the orthonormal basis then if k≠i, <k|i>=0, just because the basis is orthonormal and that's what the 'normal' part means. No need to insert the identity anywhere.
bon said:Thanks, sorry I wasn't clear. |k> and |i> are elements of the orthonormal basis. And I know that this means <k|i>=0 if k doesn't equal i. It's just that (as part of a larger calculation) I have arrived at the expression
Ʃ(j)<k|j><j|i> (where |j> is also an element of the orthnormal basis) and just wanted to check this equals <k|i>. Am I correct in thinking it does?
Thanks again
In quantum mechanics, a basis is a set of states or vectors that can be used to express or represent any other state or vector within the system. It serves as a foundation for describing the state of a quantum system and allows for mathematical calculations and predictions.
The choice of basis is important in quantum mechanics because it affects the ease and accuracy of calculations and predictions. Different bases may be more suitable for describing certain physical systems, and choosing the appropriate basis can simplify the mathematical representation of the system.
In quantum mechanics, the basis is determined by the physical properties of the system being studied. For example, in a system with discrete energy levels, the basis may be determined by the energy states of the system. In a system with continuous variables, the basis may be determined by position or momentum states.
No, a single state or vector cannot be used as a basis in quantum mechanics. A basis must consist of a set of states or vectors that are linearly independent, meaning they cannot be expressed as a combination of each other. This allows for a unique representation of any state within the system.
The Heisenberg uncertainty principle states that it is impossible to know both the exact position and momentum of a particle at the same time. This is related to the concept of basis in quantum mechanics because the choice of basis can affect the precision with which these properties can be measured. The more accurately one property is known, the less accurately the other can be known, and the choice of basis can impact this trade-off.