7/31/2019 A tensor

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A tensor \sigma_{ij} can be used to represent the stress (Force Per Area) that acts on a small

element of mass. If you do a search on 'stress tensor' in google, the wikipedia artical comes up

that explains this tensor and gives a picture. One dimension of the tensor (say the index 'i')

represents surfaces of a small element of mass (or sides of the cube). The other dimension (index

'j') represents the set of forces that act on each of the surfaces. For each surface there can be one

normal force (normal to the surface) and two shear forces (acting tangent to the surface). Note

that although the cube has six sides, a tensor only represents three sides because the element is

considered to be infinitesimal and the components of stress on one side of the cube are taken to

be roughly equal and opposite those on the other side of the cube.

Another use of tensors is to define a relationship between two different types of tensors

quantities. For example, the stress tensor \sigma_{ij} repesents the Force per Area acting on a

small element of mass. Another tensor called the strain tensor \epsilon_{kl} represents the shape

of a small element of mass (for example, a small element of mass may be stretched like a

rectangle, or possibly sheared like a diamond). You would expect that applying forces to a bodycauses the shape to change. In other words, there should be some relation between the stress in

the body \sigma_

{ij} (local force per area) and the strain \epsilon_{kl} (local change in shape). Often it is

assumed that the relationship between the stress and strain is linear and a fourth order tensor

E_{ijkl} called the Modulus of Elasticity is determined which relates stress to strain. The

relationship \sigma_{ij}=E_{ijkl} \epsilon_{kl} is called Hooke's Law and relates the

deformation of the mass to the associated stresses that the deformation causes. The Modulus of

Elasticity is a material property that is determined empiracally (by strestching or twisting a the

material of interest and seeing how much it deforms). For certain stress ranges, it turns out that a

linear relationship between stress and strain (represented by the fourth order tensor E_{ijkl]) ispretty good. This is referred to as the plastic region for the material. Beyond this point, the

relationship between stress and strain tends not to be linear. This region is referred to as the

elastic region the Modulus of Elasticity is not valid. Note that Hooke's law is just a complicated

version of the equation F=-kx for a spring which relates the stretch of a spring the the amount of

force that is required to stretch the spring that far.

Another common use of tensors is to relate two coordinate systems and convert vectors (or other

tensors) from one coordinate system to a new coordinate system. For example, suppose you have

a two dimensional newtonian mechanics problem F=ma represented in terms of coordinates x_1

and x_2. The problem is rather complicated in terms of x_1 and x_2, so you choose a new set of

coordinates y_1 (x_1 ,x_2 ) and y_2 (x_1 ,x_2 ) which (hopefully) simplifies the problem. The

transformation to the new coordinate system is done via the chain rule. The chain rule is often

represented in terms of the jacobian matrix which is multiplied by a vector to convert it from one

set of coordinates to a new set of coordinates. Wikipedia has an article on the chain rule and on

the jacobian which could give more clear interpretation in my opinion, but are worth looking at.

7/31/2019 A tensor

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when we are talking about states in quantum mechanics I like to think about it like this.

Let's say we have to independent particles A and B, which both can be in the states 0 and 1. Fisrt

let's think about what we meenby independent. We mean that when asked if the system is in a

state ex. |00> (both A and B in state 0), we would like to calculate the probability by

=

next we meen that the particles can be in states independent of each other that is we have the

possible states

|00>,|10>,|01>,|11>

So we start with two two dimensional statespaces lets say H_A and H_B (both with basis

|0>,|1>), we now want to construct a new state space for the joint system that are again a vectorspace (in fact a Hilbertspace), with the porperties mensioned above already built in it.

So we make a new construction called the tensor product of H_A and H_B

H = H_A tensor H_B

it can be shown that the new space has a basis given by

|0> tensor |0>, |1> tensor |0>, |0> tensor |1> and |1> tensor |1>

ok so it seems like we have got or four states |00>,|10>,|01>,|11>, but is it reasonble to identify

these states with the basis vectors of H. It is because when we have to hilbert spaces with inner

product < | >, then the tensor product of these becomes an Hilbert space witch an inner product

given by

() =

sp if we indentufy |1> tensor |0> with |10> and so on, we see that the tensor product reflect or

desired properties and is therefor a good way to work with these independent statespaces. This

can of cause be generalized to more spaces and more states in the spaces.