## PHY456H1F: Quantum Mechanics II. Lecture 1 (Taught by Prof J.E. Sipe). Review: Composite systems

Posted by peeterjoot on September 15, 2011

Peeter’s lecture notes from class. May not be entirely coherent.

# Composite systems.

This is apparently covered as a side effect in the text [1] in one of the advanced material sections. FIXME: what section?

Example, one spin one half particle and one spin one particle. We can describe either quantum mechanically, described by a pair of Hilbert spaces

of dimension

of dimension

Recall that a Hilbert space (finite or infinite dimensional) is the set of states that describe the system. There were some additional details (completeness, normalizable, integrable, …) not really covered in the physics curriculum, but available in mathematical descriptions.

We form the composite (Hilbert) space

for any ket in

where

Similarly

for any ket in

where

The composite Hilbert space has dimension

basis kets:

where

Any ket in can be written

**Direct product of kets:**

If in cannot be written as , then is said to be “entangled”.

FIXME: insert a concrete example of this, with some low dimension.

## Operators.

With operators and on the respective Hilbert spaces. We’d now like to build

If one defines

**Q:Can every operator that can be defined on the composite space have a representation of this form?**

No.

Special cases. The identity operators. Suppose that

then

### Example commutator.

Can do other operations. Example:

Let’s verify this one. Suppose that our state has the representation

so that the action on this ket from the composite operations are

Our commutator is

### Generalizations.

Can generalize to

Can also start with and seek factor spaces. If is not prime there are, in general, many ways to find factor spaces

A ket , if unentangled in the first factor space, then it will be in general entangled in a second space. Thus ket entanglement is not a property of the ket itself, but instead is intrinsically related to the space in which it is represented.

# References

[1] BR Desai. *Quantum mechanics with basic field theory*. Cambridge University Press, 2009.

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