# Quantum Mechanics Study Notes 1: basic rules

**/************ Definitions ************/**

**Math definition 1**: A linear operator X has a number of eigenvalues *x1*, *x2*, …, *xn*, and their corresponding eigenstates, |*x1*>, |*x2*>, …, |*xn*>, which satisfies:

X|*x1*> = *x1*|*x1*>

X|*x2*> = *x2*|*x2*>

…

X|*xn*> = *xn*|*xn*>

Those eigenvalues are complex numbers.

**Math definition 2**: each vector |*x*> comes with a dual vector <*x*|, and a dot product system, such that

- <
*x*|*y*> = (<*y*|*x*>)* - <
*x*|(*c*|*y*>)=(*c*<*x*|)*y*> where*c*is a complex number. - <
*x|*(|*y>+|z>*)=<*x*|*y*>+<*x*|*z*>

Because of 1, we know <*x*|*x*> is always a real number. Define it as the norm of |*x*>.

Example of such systems: |*x*> is a linear vector, <*x*| is the complex conjugate of its transpose, multiplied by any constant *c*. Any *c* will work for QM — the absolute value of <*x*|*y*>does not really matter; we just care about their relative scales. In other words, we just care about removing a dimension of freedom.

For convenience, from now on, we would go without saying that all vectors are normalized.

**Math definition 3**: <*a*|X is such a thing that, for any |*b*>, (<*a*|X)|*b*> = <*a*|(X|*b*>).

**Math definition 4**: For an operator X, X* is such a thing that, for any |*a*> and |*b*>, <*a*|X|*b*> = (<*b*|X*|*a*>)*.

**Math definition 5**: operator X is called “Hermitian” if X = X*.

**/************ Theorems ************/**

**Math theorem 1**: if |*y*> = *a*|*x*>, then <*y*|=*a**<*x*|.

Proof: take any vector |*z*>, we have *a**<*x*|*z*> = *a**(<*z*|*x*>)*=(*a*<*z*|*x*>)*=(<*z*|*y*>)*=<*y*|*z*>.

**Math theorem 2**: any vector *|a> *can be expressed as sum of eigenvectors: *|a> = a1|x1>+a2|x2>+…+an|xn>.*

Proof: (missing… )

**Math theorem 3**: Hermitian operator only has real eigenvalues.

Proof: <*a*|X|*a*> = <*a*|X*|*a*> = (<*a*|X|*a*>)*, therefore <*a*|X|*a*> is a real number for any *|a>*.

Now if |*a*> is a eigenvector, then <*a*|X|*a*>= <*a*|(*a*|*a*>)=*a*<*a*|*a*>, and <*a|a>* is a real number, therefore *a* is a real number.

**Math theorem 4**: for a hermitian X and its eigenvector |*a*>, <*a|*X = *a**<*a*|.

Proof: for any |*b*>*, *<*a|*X|*b*> = (<*b|*X*|a>)*=(<*b|*X|a>)*=(*a*<*b|*a>)*=*a**<*a*|b>.

**Math theorem 5**: if two eigenstates have different eigenvalues, then their dot product is 0.

Proof: <*xi*|X|*xj> = <xi|*(*xj|xj*>) = *xj*<*xi|xj>*

Also, <*xi*|X|*xj> = (*<*xi*|X)|*xj>* = *xi*<*xi|xj>.*

Therefore, *xi* = *xj *or <*xi|xj> = *0.

**/************ QM Rules ************/**

**QM Rule 1**: An observable is represented by a hermitian operator X.

**QM Rule 2**: Eigenvalues {*x1*, … , *xn*} are all the possible outcomes of measurement X. It could be discrete, e.g. spin of an electron, or infinte and continuous, e.g. position of a particle.

**QM Rule 3**: A state is represented by a vector in that same linear space, e.g. |*a>*.

**QM Rule 4**: expand*|a> = a1|x1>+a2|x2>+…+an|xn>*, where *ai *are complex numbers, and|*xi*> are eigenvectors with norm 1, then the probability of measuring X on |*a*> and get *xi* outcome is *ai*ai.*

**Note on QM Rule 4**: Sum* ai*ai = *1 is guaranteed.

This is required o make *ai*ai* a plausible probability distribution. Luckily this is guaranteed by the normalization rules above: 1=<*a|a*> = Sum_*i*_*j* *ai***aj*<*xi*|*xj*> = Sum_*i* *ai***ai* <*xi*|*xj> = *Sum* ai*ai.*

However, this normalization rule is by no means required by QM. If we don’t require normalization, then the probability of getting *xi* in measurement would simply become *ai*ai<xi|xi>*/<a|a>.

**QM Rule 4 (continuous space)**: expand*|a> = S f*(x)*|x>dx*, where *f*(*x*)* *are complex numbers, and the norm of |*x*> are all 1, then the probability of measuring X on |*a*> and get *x *in range of* u, v *is *S* *f*(*x*)**f*(*x*)*dx.*