Bra-ket notation, also known as Dirac notation, is a mathematical notation used in quantum mechanics to represent quantum states and operators. It uses the symbols ⟨ and ⟩ to represent the "bra" and "ket" vectors, respectively.
In the Schrodinger Equation, the bra-ket notation is used to represent the wavefunction ψ, which describes the quantum state of a system. The bra vector ⟨ψ| represents the initial state of the system, and the ket vector |ψ⟩ represents the final state after a time t.
The expression ⟨ψ|ψ⟩ in the Schrodinger Equation represents the probability of finding the system in the state ψ. It is also known as the inner product or the overlap integral of the wavefunction with itself.
The Schrodinger Equation is solved by applying the Hamiltonian operator H to the wavefunction ψ and setting it equal to the energy E of the system. This results in the eigenvalue equation H|ψ⟩ = E|ψ⟩, which can be solved to find the wavefunction ψ and the corresponding energy levels.
Bra-ket notation simplifies the mathematical expressions in quantum mechanics and makes them easier to manipulate. It also allows for a more intuitive understanding of quantum states and operations. Additionally, it is a universal notation that can be used for any quantum system, regardless of its dimensionality or complexity.