Question

This course is a general requirement and the topic is called "QM for everyone"

In his book, Albert describes the two physical "spin" states as properties of hardness and color. So a particle may be (hard, white), (hard, black), (soft, white), or (soft, black). Please address the following question in detail.

Use both the vector space formalism and Albert’s hardness and color boxes to explicate the following statement:

“In QM, when a particle posses a definite value for one physical magnitude it means that the particle is in a superposition of another, incompatible, physical magnitude".

Answer 1

• Suppose there is one box built for measuring color, one for measuring hardness, one after the other, but never together. When electrons are fed through the box for measuring Color there are two possible exits: an aperture marked B where black electrons come out and an aperture marked W where white electrons come out. For every black electron that enters the color box, it exits B, and for every white electron, it exits W. Thus, the color of every electron can be inferred from what aperture it exits, either B or W.
• By definition, if a certain electron is measured to be black, it hasexited aperture B, and if that same electron was fed through another color box, then that electron will with certainty alsoemerge from the second color box through aperture B. The same goes for the hardness box.
• Consider this thought experiment , a series of three boxes are constructed in a row: a color box, then a hardness box, and then another color box. A single electron enters the aperture of the first color box that exits the white aperture, and then it continues and enters into the hardness box where it emerges from the soft aperture. The same electron, now observed to be white and soft, enters the third color box. No tampering or altering has occurred between or with any of the boxes linked in this experiment. Most likely, we would expect that electron to emerge again from the white aperture, thus confirming the observation of the first color box and our knowledge that the electron we initially measured is indeed still white. However, we receive unexpected results. It is black. Actually it has 50/50 chance of being Black or White. In fact, this same 50/50 split is obtained even if we had two hardness boxes and one color box (hardness box + color box + hardness box). Half are soft, the other half, hard.

This is an example of the characteristics of uncertainty and incompatibility in quantum mechanics that, if we obtain knowledge about one property of an electron, like hypothetical white color, then we cannot obtain certain knowledge of whether that same electron is hard or soft. And if we obtain knowledge that the electron is hard, we cannot obtain certain knowledge of whether that same electron is white or black. Knowledge of these two properties are said to be incompatible—that is, we can never know both properties, color and hardness, simultaneously.

Now in terms of vector space formalism, this simply means, if for a given quantum system if we know one of the conjugate variables(e.g. momentum or position), we can't be sure about the value of the second variable.

This is actually known as Heisenberg’s principle of uncertainty (ΔxΔp>=h) which refers to a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such   as   position   and   momentum   (i.e., hardness   and color in   our example), can be simultaneously known