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DuckAmuck
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Is |x+y> = |x> + |y> ?
Thank you.
Thank you.
DuckAmuck said:Is |x+y> = |x> + |y> ?.
I see, so I can define |x+y> = |x> + |y>, but it's not implied automatically.George Jones said:The stuff inside a ket is some type of a label, and properties of a label depend the particular system, and on the type of label used.
For example, consider standard notation for the energy eigenstates of a harmonic oscillator. Then,
$$\left| 5 \right> \neq \left| 2 \right> + \left| 3 \right>.$$
DuckAmuck said:What am i missing?
No, if you define it that way you will only cause confusion for others.DuckAmuck said:I see, so I can define |x+y> = |x> + |y>, but it's not implied automatically.
The problem with your expression has nothing to do with Quantum Mechanics specifically, but rather results from a confusion between mathematical objects, on the one hand, and labels for mathematical objects, on the other. Your expression is analogous to the following classical expression relating the velocities of two people: $$\vec {V}_{Jack} + \vec {V}_{Jill} = \vec {V}_{Jack + Jill}$$ There are situations where a family of labels can indeed have a useful algebraic structure (e.g. time ##t##), but the labels used in Dirac kets in this way are not among them.DuckAmuck said:Is |x+y> = |x> + |y> ?
"Braket notation" is a mathematical notation used to describe quantum states and operators. It consists of two symbols, < and >, which represent the "bra" and "ket" vectors, respectively. The notation is commonly used in quantum mechanics and quantum computing.
In quantum mechanics, "Braket notation" is used to represent quantum states and operators. States are represented by kets, which are column vectors, and operators are represented by bras, which are row vectors. The inner product of a bra and ket represents the probability amplitude of measuring the state represented by the ket with the operator represented by the bra.
The < and > symbols in "Braket notation" represent the "bra" and "ket" vectors, respectively. These vectors are used to represent quantum states and operators in a concise and consistent manner. The left angle bracket represents a row vector, and the right angle bracket represents a column vector.
"Braket notation" simplifies calculations in quantum mechanics by providing a concise and consistent way to represent quantum states and operators. It also allows for easy manipulation of states and operators using mathematical operations such as addition, multiplication, and inner products.
Yes, "Braket notation" can be used in other fields besides quantum mechanics. It has been adapted for use in quantum information science, quantum cryptography, and quantum computing. It can also be used in linear algebra and functional analysis to represent vectors and operators.