#### Abstract Title

Differential Equations and Algebraic Operators in Quantum Mechanical Systems

#### Abstract

Quantum Mechanics is an incredibly difficult subject to understand, even to physicists. To obtain solutions in the field of Quantum Physics, and hence to be able to make predictions in the physical world, there are two different mathematical methods that are utilized. However, it is not obvious that these two methods are actually equivalent, since they are incredibly different both visually and mathematically. One way of tackling tough problems in Quantum Mechanics, is by utilizing the more intuitive branches of math which are Calculus and Differential Equations. Although it helps understand the problem more, it is significantly more time consuming than using the Linear Algebra approach, which involves matrices and operators. Again, it is not obvious that these methods are actually the same, so I set out to prove it once and for all. This past summer, I proved that these two methods are indeed equivalent by solving the Hydrogen atom wavefunction using both of these methods, proving by construction that these two methods are indeed equivalent.

#### Modified Abstract

Quantum Mechanics is an incredibly difficult subject to understand, even to physicists. To obtain solutions in the field of Quantum Physics, and hence to be able to make predictions in the physical world, there are two different mathematical methods that are utilized. However, it is not obvious that these two methods are actually equivalent, since they are incredibly different both visually and mathematically. The two alternative strategies Physicists use to attack Quantum Mechanics, are Calculus (and Differential Equations) then also Linear Algebra. This past summer, I set out to prove it once and for all. I proved that these two methods are indeed equivalent by solving the Hydrogen atom wavefunction using both of these methods, proving by construction that these two methods are indeed equivalent.

#### Research Category

Physics/Chemisty/Liquid Crystal

#### Mentor #1 Information

Dr. Declan

Keane

#### Mentor #2 Information

Dr. Brett

Ellman

#### Presentation Format

Poster

#### Start Date

April 2019

#### Research Area

Physics

Differential Equations and Algebraic Operators in Quantum Mechanical Systems

Quantum Mechanics is an incredibly difficult subject to understand, even to physicists. To obtain solutions in the field of Quantum Physics, and hence to be able to make predictions in the physical world, there are two different mathematical methods that are utilized. However, it is not obvious that these two methods are actually equivalent, since they are incredibly different both visually and mathematically. One way of tackling tough problems in Quantum Mechanics, is by utilizing the more intuitive branches of math which are Calculus and Differential Equations. Although it helps understand the problem more, it is significantly more time consuming than using the Linear Algebra approach, which involves matrices and operators. Again, it is not obvious that these methods are actually the same, so I set out to prove it once and for all. This past summer, I proved that these two methods are indeed equivalent by solving the Hydrogen atom wavefunction using both of these methods, proving by construction that these two methods are indeed equivalent.