QHO Solutions: What is Imaginary Part?

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In summary, the imaginary part, also known as the imaginary component or imaginary number, is a mathematical concept that includes the square root of negative numbers and is represented by the letter 'i'. It is an important part of complex numbers and is used in various fields such as engineering, physics, and mathematics to solve problems involving real and imaginary quantities. The imaginary part is crucial in understanding complex equations and has many practical applications, making it an essential concept to grasp in the study of mathematics.
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The solutions, in the position basis, of the Schrodinger Equation for the Quantum Harmonic Oscillator are a family of functions based on the Hermite Polynomials. The Wikipedia link for this subject is http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator .

But this Wikipedia article and most of what I have read on this topic only descirbes the REAL part of this family of solutions. What is, or where can I read about, the IMAGINARY part of these wave functions?

As always, thanks in advance.
 
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There is a theorem (see Sakurai and Napolitano, Modern QM, Second Edition) that for a time-reversal-independent Hamiltonian (such as the one under consideration here), non-denegerate energy eigenfunctions of a spinless particle can always be chosen to be real. Then choosing them to be purely real, rather than having a global phase factor, is merely an extremely sensible convention.
 

FAQ: QHO Solutions: What is Imaginary Part?

What is the imaginary part of a complex number?

The imaginary part of a complex number is the part of the number that includes the imaginary unit, i, multiplied by a real number. It is typically denoted as bi, where b is the imaginary coefficient. It is used in conjunction with the real part to form a complex number, which is represented in the form a + bi, where a is the real part and bi is the imaginary part.

What is the purpose of the imaginary part in complex numbers?

The imaginary part in complex numbers is essential for representing numbers that cannot be expressed as real numbers. It helps in solving mathematical problems and equations that involve square roots of negative numbers. Additionally, the imaginary part allows for the representation of complex physical quantities, such as alternating currents, in the field of physics.

How is the imaginary part related to the real part in a complex number?

The imaginary part and the real part in a complex number are independent of each other, meaning that they represent two separate components of the number. However, they are both necessary for forming a complex number and cannot be separated. The real part represents the horizontal axis on the complex plane, while the imaginary part represents the vertical axis. Together, they form the coordinates of the complex number on the plane.

What is the difference between the imaginary part and the imaginary unit?

The imaginary part is a component of a complex number, while the imaginary unit is a mathematical concept that represents the square root of -1. The imaginary unit, denoted as i, is used to form the imaginary part of a complex number, but it is not the same as the imaginary part itself. The imaginary part is a coefficient that is multiplied by the imaginary unit to form a complex number.

What are some real-life applications of the imaginary part?

The imaginary part, along with the real part, is used in various fields such as engineering, physics, and mathematics. In engineering, it is used to represent and solve problems involving alternating currents. In physics, the imaginary part is used to represent complex physical quantities such as impedance and electric fields. In mathematics, it is used in solving equations involving square roots of negative numbers and in representing complex numbers on the complex plane.

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