Ket Notation - Effects of the Projection Operator

[Questioneer]

Ket Notation -- Effects of the Projection Operator

Homework Statement

From Sakurai's Modern Quantum Mechanics (Revised Edition), it is just deriving equation 1.3.12.

Homework Equations

[tex] \begin{eqnarray*}\langle \alpha |\cdot (\sum_{a'}^N |a'\rangle \langle a'|) \cdot|\alpha \rangle \end{eqnarray*} [/tex]

The Attempt at a Solution

The summation can be moved to the left, so everything is being summed from a' to N, but does an alpha bra inner product with a' (or <α|a'>) does the sum of this from all a' to N equal Ʃ<a'|α>? maybe this is simple and I just can't see it?

 

[fzero]

Check (1.2.12), which is a fundamental property of the inner product. That property holds even when the bras and kets correspond to different bases.

 

[Questioneer]

Check (1.2.12), which is a fundamental property of the inner product. That property holds even when the bras and kets correspond to different bases.

So, is it in this particular case that they are equal because we are considering the eigenkets of A, a hermitian operator? Because these are the eigenkets of A, does that mean that all operators on it are real? Even though it is the operator <α| acting on it and not A?

 

[fzero]

So, is it in this particular case that they are equal because we are considering the eigenkets of A, a hermitian operator? Because these are the eigenkets of A, does that mean that all operators on it are real? Even though it is the operator <α| acting on it and not A?

There's no assumption here that [itex]\langle \alpha | a'\rangle[/itex] is real, just that [itex]\langle \alpha | a'\rangle = \langle a'| \alpha\rangle^*.[/itex] That is why the absolute value appears in the formula.

 

[paranormal]

Ket notation, also known as bra-ket notation, is a powerful tool used in quantum mechanics to represent and manipulate quantum states. The projection operator, denoted by |a'><a|, is a fundamental concept in quantum mechanics and is used to project a quantum state onto a specific subspace. This operator has important effects on the resulting state and can be represented in ket notation as |a><a|.

In the given equation, the summation is being performed over all possible states a' to N. This means that the resulting state will be a combination of all these states, weighted by their respective inner product with the state |α>. In other words, the projection operator is projecting the state |α> onto the subspace spanned by the states |a'>, and the resulting state is a linear combination of these projected states. This is a powerful tool in quantum mechanics, as it allows us to manipulate and study quantum states in a more intuitive way.

Moving the summation to the left does not change the overall result, as it is simply a rearrangement of terms. However, it is important to note that the projection operator is a Hermitian operator, meaning that its transpose is equal to its complex conjugate. This property ensures that the resulting state will also be normalized, preserving the probabilistic interpretation of quantum mechanics.

In conclusion, the projection operator in ket notation has important effects on the resulting state, allowing us to project and manipulate quantum states in a more intuitive way. It is a fundamental concept in quantum mechanics and is crucial for understanding and solving problems in this field.