string theory

Examples of string theory in the following topics:

• Implications of Quantum Mechanics

  • Quantum mechanics has also strongly influenced string theory.

• de Broglie and the Bohr Model

  • When an electron is bound to an atom, its wavelength must fit into a small space, something like a standing wave on a string.
  • The new theory was proposed by Werner Heisenberg.
  • By different reasoning, another form of the same theory, wave mechanics, was discovered independently by Austrian physicist Erwin Schrödinger.
  • (a) Waves on a string have a wavelength related to the length of the string, allowing them to interfere constructively.
  • (b) If we imagine the string bent into a closed circle, we get a rough idea of how electrons in circular orbits can interfere constructively.

• The Speed of a Wave on a String

  • Imagine you are holding one end of a string, and the other end is secured and the string is pulled tight.
  • Now, if you were to flick the string either up and down.
  • Another example of waves on strings are of the waves on vibrating strings, such as in musical instruments.
  • Pianos and guitars both use vibrating strings to produce music.
  • The speed of a wave on this kind of string is proportional to the square root of the tension in the string and inversely proportional to the square root of the linear density of the string:$v=\sqrt{\frac{T}{\mu}}$

• Sound Production: Vibrating String and Air Columns

  • A vibrating string or air column can both create music and have unique properties.
  • There are many instruments that produce sound based on strings.
  • These sounds are produced by standing waves in the strings.
  • Vibration, standing waves in a string.
  • Calculate the frequency of the sound wave produced by the string and a column of air

• Standing Waves and Resonance

  • Standing waves are found on the strings of musical instruments and are due to reflections of waves from the ends of the string. shows seven standing waves that can be created on a string that is fixed at both ends.
  • The fixed ends of strings must be nodes, too, because the string cannot move there.
  • Standing waves on strings have a frequency that is related to the propagation speed vw of the disturbance on the string.
  • The wavelength λ is determined by the distance between the points where the string is fixed in place.
  • Standing waves in a string, the fundamental mode and the first six overtones.

• Standing Waves on a String

  • When you pluck a string, it appears to vibrate.
  • This is the basis not only for a guitar, but any other string instrument.
  • There are two scenarios of waves in strings: the string is fixed at both ends, or the string is fixed at one end and free at the other.
  • A transverse wave will move along the string until it reaches the other end.
  • When we observe a standing wave on strings, it appears the wave is not moving but standing still.

• Wave Nature of Matter Causes Quantization

  • We will look at a basic string "instrument" (a string pulled tight and fixed at both ends).
  • If a string was free and not attached to anything, we know that it could oscillate at any driven frequency.
  • However, the string in this example (with fixed ends and specific length) can only produce a very specific set of pitches because only waves of a certain wavelength can "fit" on the string of a given length with fixed ends.
  • Once the string becomes a "bound system" with specific boundary restrictions, it allows waves with only a discrete set of frequencies.
  • Just like a free string, the matter wave of a free electron can have any wavelength, determined by its momentum.

• Harmonic Wave Functions

  • When vibrations in the string are simple harmonic motion, waves are described by harmonic wave functions.
  • We assume there is no loss of energy during transmission of wave along the string.
  • This can be approximated when the string is light and taught.
  • The amplitude of wave form remains intact through its passage along the string.
  • Each particle (or a small segment of string) vibrates in SHM.

• Reflection and Transmission

  • Consider a long string made by connecting two sub-strings with different density $\mu_1, \mu_2$ .
  • When the string is driven by an external force, partial reflection and transmission occurs as in Figure 18426.
  • We choose our coordinates such that the junction of two sub-strings is located at x=0.
  • Two strings with different density are connected and driven by an external driving force.

• Reflections

  • When transverse waves in strings meet one end, they are reflected, and when the incident wave meets the reflected wave, interference occurs.
  • Imagine you are holding one end of a string, and the other end is secured and the string is pulled tight.
  • Now, if you were to flick the string either up and down.
  • is an image of a transverse wave on a string that meets a free end.
  • When we observe standing waves on strings, it looks like the wave is not moving and standing still.